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Mathenatic, 
accountant 


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Southern  Branch 
of  the 

University  of  California 

Los  Angeles 


Form  L-1 


This  book  is  DUE  on  tne  last  aate  siampea  oeiow. 


;.^A/  4J  y  1923 

OCT  2  0  19^4 
DEC  1  5  1924 


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WAR   4 


OCT  2  1  1946 
MAR  ^     1949 

JAN  3     1957 
APR  2  2  I960 


MAR  7     iQ?c 


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SEP15  1962 


MATHEMATICS 


FOR   THE 


ACCOUNTANT 


BY 

EUGENE  R.  VINAL,  A.M. 

Professor  of  Actuarial  Mathematics  and 

Accounting 

School  of  Commerce  and  Finance 

Northeastern  College,  Boston 


SECOND   EDITION 


NEW  YORK 
BIDDLE    BUSINESS  PUBLICATIONS,   Inc. 

1921 

'3  7-/6/ 


Copyright,   1920,  by 
The  Biddle  Publishing  Company. 


Copyright,  1921,  by 
BiDOLE  Business  Publications,  Inc. 


PREFACE 

This  book  is  the  outgrowth  of  a  course  which  has  been 
given  for  several  years  in  the  School  of  Commerce  and 
Finance  of  Northeastern  College.  It  has  always  been  the 
policy  of  the  School  to  give  its  students  a  thorough  train- 
ing in  all  phases  of  accountancy,  and  annuity  studies  have 
been  considered  a  part  of  the  training  for  that  profession. 
While  the  course  is  forming  in  the  early  part  of  the  year, 
it  has  seemed  well  to  treat  some  elementary  subjects 
which  frequently  appear  in  C.  P.  A.  examinations,  such 
as  averaging  of  accounts  and  foreign  exchange,  and  at 
the  end  of  the  year  it  has  seemed  practical  to  include  a 
few  lectures  on  the  slide  rule 

A  new  impetus  was  given  to  tne  work  by  an  editorial 
which  appeared  in  the  ''Journal  of  Accountancy"  for 
August,  1918.  By  that  editorial  the  American  Institute  of 
Accountants  committed  itself  definitely  to  the  policy  of 
requiring  certain  annuity  studies  of  all  candidates  for 
its  examinations.  The  editorial  read  in  part  as  follows: 
"The  scope  of  the  examination  in  Actuarial  Science  is  to 
include  certain  problems  relative  to  interest  and  annuities, 
certain  sinking  funds,  loans  repayable  by  instalments, 
and  so  forth,  and  also  the  construction  and  use  of  tables 
relating  thereto.  In  other  words,  the  candidate  will  be 
expected  to  answer  questions  based  upon  a  knowledge 
which  will  have  been  obtained  in  the  study  of  algebra. 


PREFACE 

Any  candidate  who  has  an  intelligent  conception  of  algebra 
should  have  no  difficulty  in  answering  the  questions  which 
will  be  propounded  in  actuarial  science.    .    .    . 

"There  is  a  great  deal  to  be  said  in  favor  of  the  inclusion 
of  actuarial  problems  and  we  believe  that  the  need  for 
knowledge  of  this  kind  will  increase  as  time  goes  on. 
Heretofore  there  have  been  many  accountants  who  have 
had  practically  no  need  to  exercise  any  knowledge  which 
they  have  possessed  of  actuarial  matters,  but  with  the 
growth  of  accounting  work  and  the  broadening  of  its 
scope,  there  must  be  many  problems  which  can  be  solved 
at  a  great  saving  of  time  and  effort  if  the  accountant  be 
able  to  deal  with  them  with  the  advantage  of  actuarial 
knowledge." 

This  requirement  seems  to  imply  a  general  study  of  com- 
pound interest  and  its  application  to  those  problems  which 
are  commonly  solved  by  simple  interest.  It  is  axiomatic 
that  every  dollar  in  the  business  world  is  at  work;  it  is 
earning  other  dollars  either  for  its  owner  or  for  someone 
else.  At  the  end  of  every  fiscal  period  all  these  earnings, 
after  deduction  of  expenses,  should  become  capital,  and 
the  same  or  a  larger  income  rate  should  be  earned  on  this 
increased  capital.  Such  studies  are  compound  interest  and 
present  worth,  with  the  various  aspects  of  interest  rates; 
annuities,  immediate,  due  or  deferred;  their  amount  and 
present  worth;  sinking  funds  and  various  related  prob- 
lems in  valuation  of  assets ;  amortization ;  bond  valuation 
in  its  many  aspects. 

It  is  obvious  that  the  accountant  has  no  time  for  alge- 
braic studies,  unless  he  has  attended  a  good  high  school  or 
college.  To  prepare  a  book  for  those  who  have  not  had 
these  advantages  and  who  nevertheless  are  promising 
students  of  accountancy,  all  the  work  must  be  reduced  to 


PREFACE 

a  basis  of  arithmetic  and  common  sense.  Formulae  there 
must  be,  and  they  are  numerous  and  sometimes  com- 
plicated; yet  there  is  no  reason  why  any  man  who  is 
fitted  mentally  for  high  grade  accounting  practise  should 
have  difficulty  with  any  formulae  which  are  germane  to 
his  profession.  The  interest  rates  are  so  varied  and  so 
seldom  common  enough  to  be  included  in  annuity  tables 
that  a  study  of  logarithms  is  absolutely  necessary.  This 
study  should  be  pursued  no  further  than  is  demanded  by 
the  formulae  for  compound  amount  and  the  simpler 
formulae  for  the  time  and  rate  involved  in  a  transaction. 
In  short,  the  difficulty  has  been  to  take  several  highly 
technical  books  on  actuarial  science  and  reduce  them  to 
a  content  and  method  which  will  give  the  accountant  a 
maximum  of  training  in  a  minimum  of  time.  Some 
actuarial  works  are  suitable  for  the  student  who  knows 
some  algebra  and  likes  to  read  mathematics.  Such  are 
Todhunter's  ''Textbook  of  the  Institute  of  Actuaries, 
Book  I",  King's  'Theory  of  Finance",  and  ]\Iackenzie's 
"Interest  and  Bond  Values".  For  the  student  who  knew 
algebra  once  and  desires  to  renew  his  knowledge  there  are 
the  Went  worth-Smith  "Commercial  Algebra,  Book  2", 
and  Skinner's  "Mathematical  Theory  of  Investment". 
For  the  student  who  has  no  algebra  there  is  only  the 
Sprague-Perrine  "Accountancy  of  Investment",  which  is 
written  from  the  standpoint  of  the  savings  bank  man  or 
trustee,  and  does  not  treat  some  important  phases  of 
general  aecounting  knowledge.  These  are  all  excellent 
books,  and  have  been  of  great  service  to  the  author. 

The  most  convenient  tables  for  the  student  are  those  in 
Skinner's  "Mathematical  Theory  of  Investment",  and 
reprinted  separately  by  the  publishers,  Ginn  and  Com- 
pany. 


PREFACE 

This  book,  then,  is  for  the  use  of  the  accountant  who 
desires  to  prepare  for  the  examinations  of  the  American 
Institute  oi  Accountants  or  for  the  Ceitified  Pubhc 
Accountants'  examinations  of  the  various  states;  for  the 
accountant  who  will  be  required  to  handle  sinking  fund 
or  bond  accounts  in  a  scientific  way;  for  the  accountant 
who  may  be  called  upon  to  audit  the  accounts  of  an 
insurance  company,  savings  bank,  brokerage  house, 
or  any  concern  which  operates  its  bond  accounts  scien- 
tifically ;  and  finally  for  the  accountant  who  believes  that 
the  broadest  training  is  best  for  the  professional  man. 

The  book  ends  with  a  chapter  in  which  the  fundamentals 
of  actuarial  science  are  treated  in  a  different  manner,  as 
a  quiz  for  candidates  for  examinations.  In  this  chapter 
the  treatment  is  wholly  arithmetical,  and  no  logarithmic 
or  algebraic  knowledge  is  required.  Problems  from  the 
recent  examinations  of  the  American  Institute  of  Account- 
ants and  from  some  recent  C.  P.  A.  examinations  are 
solved  and  discussed.  Since  no  books  or  tables  are  allowed 
in  any  of  these  examinations,  the  rules  are  presented  in 
such  form  as  makes  them  easy  to  remember  and  apply. 

Inasmuch  as  this  book  is  written  for  the  benefit  of 
accountants,  their  co-operation  is  requested  and  will  be 
gratefully  accepted.  Any  suggestions  as  to  content, 
method,  or  problems  will  be  acknowledged  and  con- 
sidered seriously.  Any  formulae  which  accountants  have 
found  useful  are  particularly  welcome.  It  is  planned  that 
if  the  book  goes  through  a  revision  a  collection  of  formulae 
shall  appear  as  a  separate  chapter,  a  reference  list  such 
as  engineers  find  so  useful.  Any  C.  P.  A.  problems  will 
also  be  especially  welcome,  because  the  aspects  of  the 
subject  which  appear  important  to  examiners  are  so  varied 
that  the  most  complete  collection  which  can  be  put  to- 


PREFACE 

gether  will  be  none  too  complete  for  the  prospective 
candidate  to  study. 

In  closing,  I  wish  to  express  my  sincere  appreciation  of 
the  help  and  inspiration  I  have  received  from  Professor 
Charles  F.  Rittenhouse,  C.  P.  A.  He  first  brought  the 
subject  to  my  attention  as  suitable  matter  for  a  college 
course,  and  has  in  many  ways  inspired  the  writing  of 
this  book. 

EUGENE  R.  VINAL. 

Boston,  Massachusetts. 


ix 


CONTENTS 

Chapter  Page 

I  Preliminary  Suggestions 3 

Table  of  Multiples. 
Contracted  Multiplication. 
Contracted  Division. 
The  Number  of  Decimal  Places  Neces- 
sary for  Accuracy. 
Problems. 


II  Simple  Interest 9 

Principal. 

Rate. 

Frequency. 

Time. 

Common  Time. 

Exact  Time. 

Bankers'  Time. 

Exact  Interest. 

One  Per  Cent  Method. 

Discounting  Notes. 

Interest  on  Daily  Balances. 

Problems. 

Ill  Accounts  Current 18 

Averaging  an  Account. 

Finding  the  Equated  Date  of  an  Ac- 
count. 

Rule  for  Reckoning  from  the  Focal 
Date  to  the  Equated  Date. 

Problems. 


xii  CONTENTS 

Chapter  Page 

IV  Foreign  Exchange  25 

Par  of  Exchange  between  the  United 
States  and  Other  Commercial  Na- 
tions. 

The  Rate  of  Exchange  on  London. 

The  Value  of  a  Bill  on  London. 

Exporting  Gold. 

The  Rate  of  Exchange  on  France, 
Belgium,  Switzerland,  and  Italy. 

South  American  Exchange. 

Arbitrage  of  Exchange. 

Problems. 

V  Powers  and  Roots  :  Logarithms 33 

The  Four  Laws  of  Exponents. 

Multiplication. 

Division. 

Raising  to  a  Power. 

Finding  a  Root. 
Fractional  Exponents. 
Negative  Exponents. 
The  Zero  Exponent. 
Logarithms. 

Determining  the  Characteristic. 
Logarithmic  Tables. 
Finding  a  Logarithm. 
The  Cologarithm  of  a  Number. 
Finding  the  Antilogarithm. 
Illustrative  Problems. 

VI  Compound  Interest  AND  Present  Worth        44 
Actuarial  Science. 

Compound  Interest — Its  Significance, 
for  the  Accountant. 


CONTENTS  xiii 

Chapteb  Page 

Computation  of  Compound  Amount. 

Nominal  and  Effective  Rates. 

Changing  from  Nominal  to  Effective 
Rates. 

Nominal  Rate  when  the  Effective  Rate 
Is  Given. 

Formula  for  Finding  the  Nominal 
Rate  when  the  Effective  Rate  Is 
Given. 

Present  Worth  and  Compound  Dis- 
count. 

Summary  of  the  Processes  Used  in 
Compound  Interest  Computation. 

Problems. 

VII  Annuities  :  Amount  and  Present  Worth        60 
Annuity  Defined. 

The  Amount  of  an  Ordinary  Annuity. 
Formula  for  the  Amount  of  an  Annuity : 

Effective  Rates. 
The  Present  Worth  of  an  Annuity: 

Effective  Rates. 
Three  Special  Types  of  Annuity. 
An  Annuity  Due. 
Amount  of  an  Annuity  Due. 
The  Present  Worth  of  an  Annuity  Due. 
A  Deferred  Annuity. 
A  Perpetuity. 

The  Interest  Rate  on  a  Given  Annuity. 
Problems. 

VIII  Sinking  Funds 75 

Principles  of  Sinking  Fund  Mathe- 
matics. 


xiv  CONTENTS 

Chapter  Page 

Effective  Rates. 

Problem  Involving  Logarithms. 

Problem  Involving  Annuity  Due. 

Formula  to  Determine  the  Time  re- 
quired to  Accumulate  a  Stated 
Amount. 

Problems. 

IX  Valuation  of  Assets 81 

Valuation  of  a  Plant  as  a  Whole. 

The  Rate  of  Depreciation. 

The  Composite  Life  of  a  Plant. 

The  Wearing  Value  of  an  Asset  at 
any  Time. 

The  Condition  Per  Cent  of  a  Plant  at 
any  Time. 

Other  Methods  of  Reckoning  De- 
preciation. 

Fixed  Percentage  on  a  Diminishing 
Book  Value. 

Provision  Against  the  Total  Exhaus- 
tion of  Wasting  Assets. 

Capitalization  of  Assets. 

Summary  of  Formulae  Used  in  Ac- 
counting for  Assets. 

Problems. 


X  Amortization 93 

Relation  between  Sinking  Fund  and 

Amortization  Schedules. 
Logarithmic  Solution.  •• 

Effective  Rates. 
Annuity  Due. 


CONTENTS  XV 

Chapter  Page 

The  Amount  Due  After  Any  Number 

of  Payments. 
The   Time   Required   to   Amortize   a 

Given  Debt. 
The   Increased   Cost  of   Lengthening 

the  Life  of  an  Asset. 
Problems. 

XI  Valuation  of  Bonds 101 

Bonds  and  Their  Value. 

Formula  for  Finding  the  Correct  Value 

of  a  Bond. 
Schedule  of  Amortization. 
Redemption  at  a  Premium  or  Discount. 
Valuation  with  Allowance  for  Income 

Tax. 
Valuation  between  Interest  Dates. 
Serial  Bonds. 
Redemption  of  a  Series. 
Instalment  Bonds. 
Determining  the  Income  Rate  on  a 

Bond. 
Problems. 

XII  The  Slide  Rule 123 

The  Logarithmic  Scale, 
Rules    for    Determining    the    Char- 
acteristic. 
Illustrations    in    Multiplication    and 

Division. 
How  to  Square  a  Number. 
••   Square  Root. 
Proportion. 
Problems. 


xvi  CONTENTS 

Chapter  Page 

XIII  Review  Problems 131 

Problems. 

Two  Examination  Papers. 

XIV  Problems   from   the   Examinations   of 

THE  American  Institute  of  Ac- 
countants         143 

On  Averaging  an  Account  with  an 
English  Correspondent. 

On  Compound  Interest. 

On  Present  Worth. 

On  Compound  Discount. 

On  Annuities. 

On  Deferred  Annuity. 

On  Sinking  Funds. 

On  Depreciation. 


CHAPTER  I 
PRELIMINARY  SUGGESTIONS 

This  chapter  is  a  collection  of  processes  which  will 
prove  to  be  timesavers  in  the  use  of  tables.  It  is  not  in 
any  sense  a  collection  of  short  cuts  such  as  is  found  in  the 
texts  on  arithmetic,  but  simply  a  brief  consideration  of 
methods  of  shortening  and  simplifying  the  numerous  and 
otherwise  laborious  operations  which  are  necessary  when 
tables  of  annuities  or  logarithms  have  to  be  used  exten- 
sively. It  is  obvious  that  if  an  operation  can  be  performed 
one  way  and  lead  to  a  ten-place  answer,  and  by  another 
method  will  give  only  a  four-place  answer,  that  the  second 
method  is  preferable  for  the  accountant.  These  processes 
are  called  reversed  or  contracted  multiplication,  and  con- 
tracted division.  A  brief  consideration  of  the  number  of 
decimal  places  desirable  in  any  given  computation  will 
also  be  found  helpful. 

Table  of  Multiples :  In  any  case  where  the  same  num- 
ber must  be  used  often  as  multiplicand  or  divisor,  a  table  of 
multiples  is  a  time  saver.  It  is  formed  as  follows :  Suppose 
we  are  finding  interest  at  exact  time;  that  is  we  are  using 
365  as  a  divisor: 


1 

365 

2 

730 

3 

1095 

4 

1460 

5 

1825 

6 

2190 

7 

2555 

8 

2920 

9 

3285 

10 

3650 

4  MATHEMATICS  FOR  THE  ACCOUNTANT 

On  the  first  line  set  down  the  number.  The  next  line 
is  twice  the  number.  The  third  line  is  the  sum  of  the  first 
and  second.  The  fourth  line  is  the  sum  of  the  first  and 
third;  and  so  on.  The  tenth,  which  is  the  sum  of  the  first 
and  ninth,  is  also  ten  times  the  original  number,  and  so  is 
a  proof  line.  Now  in  multiplying  or  dividing  we  can  read 
the  desired  multiples  and  quotient  figures  from  this  table 

Contracted  multiplication,  sometimes  called  reversed 
multiplication.  Suppose  we  are  to  receive  $459.73  at  the 
end  of  five  years,  and  we  wish  to  "discount"  this  amount 
to  the  present  time,  at  5%  compounding  annually.  SI. 00 
due  in  five  years  at  5%  compounding  annually  is  worth 
$.7835262  now;  this  is  called  the  present  worth.  So  the 
present  worth  of  $459.73  is  the  product  of  these  amounts. 
Form  a  table  of  multiples : 


1 

45973 

2 

91946 

3 

137919 

4 

183892 

6 

229865 

6 

275838 

7 

321811 

8 

367784 

9 

413757 

10 

459730 

The  product  will  run  to  nine  places  of  decimals  and  we 
need  only  four.  We  can  do  away  with  the  five  superfluous 
digits  by  beginning  with  the  left-hand  digit  of  the  mul- 
tiplier and  proceeding  as  follows: 


PRELIMINARY  SUGGESTIONS 

459.73 
.7835262 


321.811 

36 . 7784 

1.3792 

2299 

91 

27 

360.2103 


Locate  the  decimal  point  in  the  first  partial  product. 
This  is  usually  a  matter  of  common  sense,  rather  than 
rule.  In  this  example,  450  X  .8  =  360;  that  is,  there  are 
three  digits  at  the  left  of  the  point. 

In  the  second  partial  product  we  have  four  places  and 
are  ready  to  reject  all  digits  at  the  right  of  the  fourth.  But 
it  is  advisable  to  take  account  of  all  ''carrying"  digits.  So 
the  third  partial  product  ought  to  be  13792  rather  than 
13791.  In  all  such  multipUcation  as  this  the  right-hand 
figure  is  inaccurate,  but  the  error  is  negligible  if  we  are 
keeping  a  sufficient  number  of  decimal  places. 

Contracted  division:  The  above  problem  could  have 
been  solved  by  division.  The  compound  amount  of  $1.00 
for  five  years  at  5%  compounding  annually  is  $1.2762816. 
Therefore  the  number  of  dollars  which  will  amount  to 
459.73  will  be  the  quotient  of  459.73  divided  by  1.2762816. 
We  will  not  show  the  table  of  multiples  here. 


MATHEMATICS  FOR  THE  ACCOUNTANT 

12762816)4597300000(360 . 21047 

38288448 


76845520 
76576896 

2686240 
2552563 

133677 
127628 

6049 
5105 


944 

893 


The  division  proceeds  as  usual  uDtil  we  reach  the  decimal 
point,  after  which  we  do  not  bring  down  any  more  digits 
from  the  dividend,  but  at  each  operation  cut  off  the  right 
hand  digit  of  the  divisor. 

The  number  of  decimal  places  necessary  for  accuracy 
varies.  In  general,  the  more  the  better;  for  the  accuracy 
of  any  result  is  somewhat  leps  than  the  accuracy  of  the 
least  accurate  item. 

In  ordinary  dollars  and  cents  work,  as  in  schedules  of 
annuities  and  bonds,  all  values  ought  to  be  carried  to  four 
places,  that  is  to  hundredths  of  a  cent. 

In  logarithmic  work,  there  ought  to  be  not  less  than  six 
places,  and  seven  are  better.  Logarithms  of  the  com- 
pound interest  ratios  must  run  to  at  least  ten  places, 
because  the  rates  are  so  close  together  and  so  near  the 
beginning  of  the  number  scale  that  less  than  ten  places 
will  not  show  the  differences  accurately.  Furthermore 
there  are  so  many  cases  in  which  the  results  of  these 


PRELIMINARY  SUGGESTIONS  7 

operations  are  multiplied  by  100,000  or  more,  that  less 
than  ten  digits  will  not  give  even  approximately  correct 
results.  Even  then  it  is  best  to  test  out  all  results  by 
reversing  the  operation  or  setting  up  a  schedule,  then 
if  an  error  appeal's  it  can  be  spread  over  the  entire  life  of 
the  transaction  by  methods  which  will  be  shown  later. 

PROBLEMS  ON  CHAPTER  I 

1.  Required  the  present  worth  of  $2,283.67  due  in  10 
years  at  4%  .compounding  annually : 

(a)  Solve  by  multiplying  by  $ .  6755642,  the  present 
worth  of  $1.00  due  in  10  years  at  4%  annually. 

(b)  Solve  by  dividing  by  $1.4802443,  the  compound 
amount  of  $1.00  for  the  same  time  and  rate. 

2.  Out  of  89,032  persons  living  at  age  25,  74,173  are  alive 
at  age  45.  Find  by  division  the  probability  that  a 
person  now  aged  25  will  live  to  age  45. 

3.  An  asset  costing  $750.00  has  an  estimated  life  of  five 
years  and  scrap  value  of  $100.00.  By  one  method  of 
reckoning  depreciation  it  is  estimated  that  this  asset 
depreciates  each  year  33.17%  (or  as  a  decimal  .3317) 
of  its  value  at  the  beginning  of  that  year.  Construct 
a  schedule  showing  the  net  value  of  the  asset  year  by 
year,  as  follows: 

Cost $750.00 

Less  depreciation,  first  year — 

750  X  .3317 248.775 

Value,  beginning  of  second  year $501 .  225 

Less  depreciation— 501.225  X  .3317...  166.2563 

And  so  on. 


MATHEMATICS  FOR  THE  ACCOUNTANT 

Value  at  end  of  fifth  year  should  be  almost  exactly 
$100.00. 

A  Profit  and  Loss  Statement  shows: 

Net  Sales $167,834.29 

Cost  of  Sales $82,387.49 

Selling  Expenses 31,093.77 

G.  &:  A.  Expenses 22,798.44 

Net  Profit xx,xxx.xx 

What  per  cent  of  the  Net  Sales  is  each  of  these  items? 

Total  of  per  cents  should  of  course  be  100%. 

(Form  a  table  of  multiples  of  the  divisor;  why?) 


CHAPTER  II 
SIMPLE  INTEREST 

"Interest  is  the  increase  of  indebtedness  through  the 
lapse  of  time." — Sprague. 

Any  interest  contract  must  take  account  of  four  things: 
Principal,  the  number  of  units  originally  invested. 
Rate,  the  part  of  a  unit  of  value,  usually  a  few  hundredths, 
which  is  added  to  each  unit  of  principal  by  the  lapse  of 
one  period  of  time.  Frequency,  the  length  of  a  unit  of 
time,  measured  in  years,  months,  or  days;  weeks  are  not 
used,  nor  are  parts  of  days,  Time,  the  number  of  these 
units  of  time  during  which  the  indebtedness  continues. 

Some  of  the  above  are  usually  represented  by  letters. 
The  principal  is  P,  the  rate  is  i,  the  time  is  n.  More 
common  than  i  is  the  symbol  for  the  amount  at  the  end 
of  one  period,  1  +  i;  this  symbol  is  constantly  used  in 
investment  mathematics.  For  instance,  on  one  dollar  for 
one  year  at  3%  1  +  i  is  1.03,  which  means  that  at  the 
end  of  each  period  the  debt  has  increased  to  1.03  of  its 
size  at  the  beginning  of  the  period. 

Since  each  dollar  increases  at  the  same  rate  as  every 
other  dollar,  it  is  correct  and  usual  to  reckon  the  interest  or 
amount  of  one  dollar,  and  multiply  the  result  by  the 
number  of  dollars  stated  in  the  problem. 

There  are  three  common  methods  of  reckoning  simple 
interest.  They  differ  according  to  the  way  in  which  the 
time  is  reckoned. 

By  common  time  the  year  is  divided  into  12  months  of 
30  days  each,  regardless  of  the  calendar.     This  method 

9 


10  MATHEMATICS  FOR  THE  ACCOUNTANT 

is  unsatisfactory,  and  ought  never  to  be  used  by  any 
person  who  claims  to  aim  at  accuracy. 

By  exact  time  the  year  is  divided  into  365  days,  and  no 
account  is  made  of  months.  This  is  the  just  method  of 
reckoning  interest,  and  is  simple  enough,  especially  when 
exact  interest  tables  can  be  used. 

By  bankers'  time  the  methods  are  combined.  The  exact 
number  of  days  is  computed,  and  the  result  is  divided 
into  groups  which  are  multiples  or  fractions  of  60  days. 
If  a  bill  is  timed  three  months  from  June  19  it  is  due 
September  19;  but  if  it  is  timed  90  days  from  June  19  it 
is  due  September  17.  There  are  many  variants  of  this 
method,  and  only  one  of  them  will  be  treated  here. 

Since  most  interest  calculations  are  for  short  periods,  the 
results  obtained  under  the  various  methods  do  not  difTer 
greatly.  Moreover,  it  is  a  simple  matter  to  change  from 
common  to  exact  time  or  from  exact  time  to  common 
time.  If  we  let  I  represent  the  interest  at  common  time 
and  I'  the  interest  at  exact  time,  we  have  two  equations: 

To  change  from  common  to  exact  P  =  I 

73. 

Rule:  From  the  ordinary  interest  subtract  1/73  of  itself. 

To  change  from  exact  to  common  1  =  1'+   :z- 

72 

Rule :  To  the  exact  interest  add  1  /72  of  itself. 

Exact  interest:  To  find  the  exact  interest  on  $382.00  at 
5%  from  May  11  to  October  1.  First  count  the  days, 
143.  Then  reckon  one  yeai''s  interest,  $19.10.  We 
have  now  to  divide  19.10  by  366  and  multiply  by  143; 
but  it  is  easier  and  more  accurate  to  multiply  first. 

One  year's  interest 19.10 

times  143 2,731.30 

divided  by  366 7.48 


SIMPLE   INTEREST  11 

In  such  computation  a  table  of  the  multiples  of  365  is 
useful,  and  also  a  table  of  the  time  between  dates,  or 
a  calendar  showing  the  number  of  each  day  in  the  year. 

One  percent  method:  Interest  by  bankers'  time  is 
usually  figured  by  some  variation  of  the  one  percent 
method;  when  the  rate  is  6%  it  is  usually  called  the 
60-day  or  200-month  method. 

At  6%  one  year's  interest  on  $1.00  is  six  cents;  therefore 
an  interest  of  one  cent  will  be  earned  in  1/6  of  a  year, 
which  by  this  method  is  60  days.  In  general,  any  prin- 
cipal at  6%  will  earn  1%  of  itself  in  60  days.  And  since 
6  days  is  1/10  of  60  days,  any  principal  will  earn  l/lO 
of  1%  of  itself  in  six  days. 

Rule :  At  6%,  pointing  off  two  places  in  the  principal 
gives  the  interest  for  60  days;  pointing  off  three  places 
gives  the  interest  for  six  days. 

Any  period  of  time  can  be  divided  into  multiples  and 
fractions  of  60  days,  as  follows: 

120  days  is  twice  60  days. 
180  days  is  3  times  60  days,  and  so  on. 
30  days  is  1/2  of  60  days. 
20  days  is  l/3  of  60  days. 
15  days  is  l/4  of  60  days. 
12  days  is  l/5  of  60  days. 
10  days  is  l/6    of  60  days. 

6  days  is  l/ 10  of  60  days. 

3  days  is  l/2    of    6  days. 

2  days  is  1/3    of    6  days. 

1  day   is  l/6    of    6  days. 

It  is  often  convenient  to  build  in  other  ways: 

15  days  is  l/2    of  30  days. 

2  days  is  1/10  of  20  days,  and  so  on. 


12  MATHEMATICS  FOR  THE  ACCOUNTANT 

Problem:  Find  the  interest  on  $125.00  from  April  18  to 
August  3  at  6%,  bankers'  time. 
The  number  of  days  is: 

April 12  days 

May 31  days 

June 30  days 

July 31  days 

August 3  days 

Total 107  days 

Divide  the  time  as  follows: 

60  days'  interest  $1.25      (point  off  2  places) 

30     "  "  .625    (1/2    of  60) 

10     "  "  .2083  (1/3    of  30) 

6     "  "  .125    (1/10  of  60) 

1     «  «  .0208  (1/10  of  10) 

Total S2.2291 

or,  rounding  up .   2.23 

If  the  rate  is  not  6%  it  is  usually  easier  to  find  at  6%  and 
adjust.  For  instance,  to  find  at  5%,  after  finding  at  6% 
divide  by  6  and  subtract.  It  is  possible,  however,  to 
work  out  special  methods  for  some  rates : 

At  5%,      pointing  off  2  places  gives  interest  for    72  days 
At  4%,  "  "  "  "  90     " 

At4H%,        "  "         "  ''  80     " 

At  3%,  "  "  "  ''  120     " 

No  such  device  can  be  used  for  odd  rates,  such  as  6)^%, 
4/^%,  and  so  forth. 

Problem:  What  is  119  days'  interest  on  $147.35  at  5%, 
bankers'  time? 


SIMPLE  INTEREST  13 

72  days'  interest  $1.4735  (point  off  2  places) 

24      «  "  .4913  (1/3  of  72) 

18      "  «  .3683  (1/4  of  72) 

4      «  «  .0818  (1/6  of  24) 

1  ''  "  .0204  (1/4  of  4) 
119                           $2'4353 

or  $2.44 

Discounting  notes :  Find  the  net  proceeds  of  a  90-day 
note  for  $350.00,  dated  May  19,  interest  at  5%,  discounted 
June  2  at  6%. 

Face $350.00 

Add  90  days'  interest  at  5% 4.38 

Value  at  maturity $354.38 

Due  date,  90  days  from  May  19  is  Aug.  17 
Term  of  discount,  from  June  2  to  Aug.  17 
is  76  days 

Discount,  76  days  at  6%  on  $354.38 4.49 

Net  proceeds $349.87 

Interest  on  daily  balances:  Many  banks  advertise  in- 
terest on  daily  balances  of  checking  accounts  at  a  nominal 
rate.  One  method  of  computing  this  interest  is  the  follow- 
ing: Each  day's  balance  is  made  out,  and  at  the  end  of  the 
month  these  balances  are  totalled  and  one  day's  interest 
is  allowed  on  the  total.  The  following  problem  will  illus- 
trate the  use  of  exact  interest :    (The  rate  used  here  is  2  %) 

Deposits      Withdrawals      Balances 

March     1  600  ...  600 

2  ...  ...  600 

3  200  300  500 

4  100  ...  600 

5  300  200  700 
B  300  150  860 


14  MATHEMATICS  FOR  THE  ACCOUNTANT 


Deposits 

Withdrawals 

Balances 

7 

.  •  • 

200 

650 

8 

. . . 

•  • . 

650 

9 

. . . 

•  •  • 

650 

10 

200 

. . . 

850 

11 

300 

550 

12 

. . . 

550 

13 

600 

450 

700 

14 

300 

300 

700 

15 

500 

350 

850 

16 

... 

850 

17 

400 

300 

950 

18 

300 

150 

1100 

19 

200 

550 

750 

20 

1200 

1350 

600 

21 

300 

600 

300 

22 

500 

100 

700 

23 

. . . 

700 

24 

400 

250 

850 

25 

300 

100 

1050 

26 

300 

200 

1150 

27 

850 

300 

1700 

28 

400 

150 

1950 

29 

200 

. . . 

2150 

30 

2150 

31 

300 

650 

1800 
28750 

/■'s  interest 

on  $28,750 

is  $1.5753,  or 

$1.58. 

If  there  are  not  many  items  in  the  month,  the  use  of 
what  may  be  termed  * 'day-numbers"  will  shorten  the 
work.  In  this  case  a  "day-number"  is  the  number  of  days 
during  which  a  balance  remains  unchanged.  For  instance, 
if  th«  balance  on  the  seventh  of  the  month  is  $500.00,  and 


SIMPLE  INTEREST  15 

there  is  no  change  until  the  fifteenth,  there  has  been  an  un- 
changed balance  of  $500.00  for  eight  days,  which  is 
obviously  equivalent  to  a  balance  of  eight  times  $500.00 
for  one  day. 

Problem :   Find  the  interest  at  2%,  exact  time,  on  the 
following  account: 
Debits:  March  2,  $500.00;  March  11,  $450.00;  March  19, 

$600.00;  March  27,  $300.00. 
Credits :  March  8,  $300.00;  March  14,  $200.00;  March  28, 
$500.00. 

Day- 
Date      Debit      Credit     Balance   Numbers"  Products 
2  500  ...  500 

8  ...  300  200 

11  450  ...  650 

14  ...  200  450 

19  600  ...  1050 

27  300  . .  .         1350 

28  ...  500  850 

Total 20100 

One  day's  interest  on  $20,100  is  $1,1014,  or  $1.16. 

A  better  method,  involving  a  different  use  of  'May- 
numbers,"  will  be  demonstrated  in  the  next  chapter. 

PROBLEMS  ON  CHAPTER  II 

5.  Find  the  interest  on  a  3H%  Liberty  Bond  for  $50.00 
from  June  16  to  December  15. 

6.  Find  the  interest  at  6%,  bankers'  time,  on  $386.55 
from  June  17  to  October  12. 

7.  Find  the  interest  at  4  3^  %,  bankers'  time,  on  $1 ,200.00 
from  July  4  to  December  25. 

8.  A  note  for  $500.00,  three  months,  is  dated  March  9, 
interest  at  5%.  It  is  discounted  May  17  at  6%.  Find 
the  net  proceeds. 


6 

3000 

3 

600 

3 

1950 

5 

2250 

8 

8400 

1 

1350 

3 

2550 

16  MATHEMATICS  FOR  THE  ACCOUNTANT 

9.     Find  the  interest  on  daily  balances  for  the  month  on 
the  following  checking  account  at  2%,  exact  time. 


Deposits 

Checks 

1 

3000 

2000 

2 

800 

500 

3 

1200 

800 

4 

560 

210 

5 

. . . 

. . . 

6 

380 

160 

7 

650 

200 

8 

1340 

620 

9 

730 

400 

10 

520 

260 

11 

1630 

1250 

12 

. . . 

13 

870 

1420 

14 

1130 

825 

15 

760 

385 

16 

920 

440 

17 

1420 

675 

IS 

1430 

1260 

19 

.... 

20 

1165 

1530 

21 

680 

445 

22 

1480 

1260 

23 

260 

420 

24 

650 

725 

25 

1180 

855 

26 

.... 

27 

380 

1560 

28 

640 

210 

29 

360 

620 

30 

1245 

1135 

31 

680 

320 

SIMPLE  INTERESI^  17 

10.  Find  the  interest  due  January  31,  at  2)/^%,  exact 
time,  on  these  daily  balances  for  the  month  of 
January : 

Debits:    3rd,  $6,800;  7th,  $4,500;  22nd,  $5,250; 
29th,  $1,850. 

Credits:    6th,  $4,250;  11th,  $5,300;  20th,  $200; 
30th.  $1,200. 


CHAPTER  III 
ACCOUNTS  CURRENT 

Under  the  title  of  this  chapter  are  included  two  types  of 
problem.  The  first  is  the  averaging  of  an  account;  that 
is,  the  statement  of  the  balance  as  of  a  certain  date,  with 
interest  charged  and  credited  at  a  certain  rate.  The 
second  is  the  finding  of  the  equated  date;  that  is,  the  date 
on  which  the  balance  of  the  account  is  at  a  minimum: 
when  the  account  can  be  settled  with  a  minimum  payment 
of  interest. 

Averaging  an  account: 

Problem:  Find  the  balance  of  the  following  account  as 
of  July  1,  interest  at  5%  bankers'  time: 

Debits:  April  27,  30-day  note  without  interest,  $500.00; 

May  18,  60-day  note  without  interest,  $600.00; 

June  22,  cash,  $1,200.00. 
Credits:  June  4,  cash,  $500.00; 
June  18,  cash,  $200.00. 

Of  course  the  30-day  note  must  be  dated  from  maturity. 
May  27;  and  the  60-day  note  from  July  17,  which  means 
that  when  we  reckon  interest  we  must  deduct  16  days' 
''discount"  for  this  item. 

There  are  two  methods  of  solution  which  give  the  same 
result.  We  may  reckon  interest  on  each  item  and  balance 
the  account  as  usual.  Or  we  may  simplify  the  work  by 
using  day-numbers. 

18 


ACCOUNTS  CURRENT 


19 


First  solution: 

Debit 

Amount 

Due 

Time 

Interest 

$500.00 

May   27 

35 

2.4304 

600.00 

July    17 

16d 

1.3333d 

1,200.00 

June  22 

9 

1.50 

$2,300.00 

2.5971 

Credit 

Amount 

Due 

Time 

Interest 

$500.00 

June     4 

27 

1.875 

200.00 

June   18 

13 

.3611 

$700.00  2.2361 

Balance  of  principal  due $1,600.00 

Balance  of  interest  due .3611 


Total  $1,600.36 

Second  solution :  In  this  solution  the  day-number  of  each 
item  is  the  number  of  days  to  the  balancing  date;  in  other 
words,  the  time  as  in  the  first  solution.  In  short  form  the 
solution  is: 

500  X  35    17,500  500  X  27     13,500 

600  X  16d    9,600  deduct      200  X  13       2,600 
1,200  X    9    10,800 

2,300  18,700  700  16,100 

Debit  balance  of  day-numbers,  2,600;  one  day's  interest 
at  5%  is  .3611,  agreeing  with  the  result  of  the  first 
solution. 

When  such  a  statement  is  made  out  on  the  exact  interest 
basis,  the  advantage  of  using  day-numbers  is  obvious. 
Under  such  conditions  the  reckoning  of  a  large  number 
of  interest  items  would  be  a  waste  of  time.  By  the  day- 
number  method  we  find  only  one  interest  item,  yet  the 
result  is  the  same  to  the  last  decimal  place. 


20  MATHEMATICS  FOR  THE  ACCOUNTANT 

Finding  the  equated  date  of  an  account:  This  has  been 
defined  as  the  date  on  which  the  balance  of  the  account  is 
at  a  minimum.  Evidently  that  is  the  date  on  which  the 
interest  is  at  a  minimum.  Consider  any  series  of  trans- 
actions. As  goods  are  purchased,  interest  on  the  in- 
debtedness begins;  as  payments  are  made,  the  indebted- 
ness is  lessened.  Between  the  purchase  and  the  payment 
there  is  a  date  on  which  the  interest  on  the  debt  and  the 
discount  on  the  payment  equalize  each  other.  A  purchase 
of  $500.00  on  July  11  and  a  payment  of  $500.00  on  August 
6  would  thus  equalize  each  other  on  July  24,  the  date 
half  way  between.  Generally,  however,  there  are  many 
items  both  debit  and  credit,  and  the  process  of  finding 
this  neutral  date  is  complicated.  It  is  necessary  to 
assume  some  date  as  a  balancing  date.  This  date  may 
be  any  date  whatever,  but  for  convenience  it  is  usual  to 
select  either  the  earliest  or  the  latest  date  in  the  problem. 
This  is  in  order  that  we  may  have  to  perform  only  one 
kind  of  operation ;  if  we  use  the  earliest  date,  we  discount 
all  items  to  that  date;  if  we  use  the  latest  date,  we  accumu- 
late all  items  to  that  date.  It  seems  easier  to  use  the 
latest  date,  and  the  solutions  given  will  all  be  in  that  form. 
After  we  have  balanced  the  account  as  of  this  date,  which 
is  called  the  "focal  date,"  we  can  find  how  many  days  out 
of  the  way  our  focal  date  is,  and  reckon  backward  or 
forward  to  the  equated  date. 

An  account  may  be  one-sided  or  two-sided ;  it  may  be  all 
debits  or  all  credits,  or  it  may  be  composed  of  both  debits 
and  credits.    We  will  study  a  one-sided  account  first. 

Problem:  Find  the  equated  date  of  this  account  at  6%, 

bankers'  time: 

Debits:  July  18,  $600.00;  July  20,  $200.00;  July  28,  30- 
day  note,  $2,800.00. 


ACCOUNTS  CURRENT  21 

The  latest  date  is  the  due  date  of  the  note,  August  27; 
we  will  use  that  as  our  focal  date. 


Amount 

Due 

Time 

Interest 

$600.00 

July         18 

40 

$4.00 

200.00 

July         20 

38 

1.27 

2,800.00 

August    27 

$3,600.00  $5.27 

Since  both  principal  and  interest  are  debits,  it  is  evident 
that  the  longer  the  account  remains  open  the  larger  the 
balance  will  be;  that  is,  the  equated  date  must  be  earlier 
than  the  focal  date.  A  balance  of  $3,600.00  is  accruing 
interest  at  the  rate  of  60  cents  a  dsij.  It  has  already 
accrued  $5.27.  This  indicates  that  interest  has  been 
accruing  for  practically  nine  days  (5.27  -=-  60).  Reckoning 
backward  nine  days  from  August  27,  we  find  the  equated 
date  to  be  August  18.  This  result  can  be  tested  by  balanc- 
ing the  account  as  of  August  18. 

Another  problem  will  show  the  solution  of  a  two-sided 
account,  and  also  the  use  of  day-numbers: 

Debits:    May  2,  30  days,  $1,200.00,  interest  at  5%. 

May  21,     $840.00. 

June     1,    1,200.00. 
Credits:  May  15,     $600.00. 

May  20,  $1,900.00. 
Interest  at  6%,  common  time. 
Focal  date,  June  1. 

Note  that  the  30-day  note  is  due  June  1 ;  but  tnat  the 
interest  on  it  cannot  appear  on  the  ledger  account,  and 
consequently  that  the  amount  appearing  in  the  statement 
is  the  face  of  the  note,  and  its  date  is  May  2. 
Assume  June  1  as  focal  date. 


22 


MATHEMATICS  FOR  THE  ACCOUNTANT 


Due 

Amount 

Days 

Products 

May  2 

$1,200.00 

30 

30,000 

May  21 

840.00 

11 

9,240 

June    1 

1,200.00 
$3,240.00 

Total 

45,240 

Paid 

Amount 

Days 

Products 

May  15 

$600.00 

17 

10,200 

May  20 

1,900.00 

12 

22,800 

Total 

$2,500.00 

33,000 

Balance  of  day-numbers  is  12,240;  one  day's  interest 
$2.04.  Balance  of  principal  is  $740.00;  one  day's  interest 
123/^  cents.  Quotient  16.7  or  17  days.  Again  we  have 
both  balances  on  the  same  side  of  the  account,  so  we  must 
reckon  backward  from  the  focal  date.  Seventeen  days 
from  June  1  is  May  15,  the  equated  date. 

Rule  for  reckoning  from  the  focal  date  to  the  equated 
date: 

If  balances  of  both  principal  and  interest  are  on  the 
same  side  of  the  account,  reckon  backward.  This  is 
because,  as  was  explained,  the  interest  on  the  principal 
is  continually  increasing  the  interest  already  accrued.  But 
if  one  balance  is  debit  and  the  other  balance  is  credit, 
reckon  forward.  This  is  because  the  interest  on  a  debit 
principal  will  tend  to  extinguish  a  credit  balance  of 
interest,  and  vice  versa. 

PROBLEMS  ON  CHAPTER  III 


Find  the  cash  balance  of  each  of  these  accounts: 
11:    Jan.  1,  6%,  bankers'  time: 

Debits:    Oct.   11,  $350.00,  30  days;  Oct.  22,  30 

days,  $662.50;  Nov.  19,  $2,258.00;  Dec.  11,  30 

days,  $350.00. 


ACCOUNTS  CURRENT  23 

Credits:  Nov.  18,  30-day  note  without  interest, 
$600.00;  Nov.  22,  cash,  $2,000.00 

12:    July  1,  5H%,  exact  time: 

Debits:  May  11,  $252.87  at  30  days;  May  13, 
$350.00  at  30  days;  May  18,  $250.00  at  20  days; 
June  11,  $353.83  at  10  days. 
Credits:  May  15,  30-day  note  for  $500.00,  interest 
at  5%;  June  1,  cash,  $200.00;  June  16,  cash, 
$200.00;  June  28,  cash,  $200.00. 

13:    May  1,  5%,  exact  time: 

Debits:  Feb.  27,  $300.00;  March  15,  $655.35,  30 
days;  March  19,  $225.75,  30  days;  March  24, 
$225.75;  April  3,  $625.38,  30  days. 

Credits:  March  1,  60-day  note,  with  interest  at 
5%,  $300.00;  April  1,  30-day  note,  without  in- 
terest, $1,000.00. 

14:    Jan.  1,  6%,  exact  time: 

Debits:  Oct.  5,  60  days,  $500.00;  Oct.  13,  30  days, 

$500.00;  Oct.  18,  30  days,  $350.00;  Nov.  8,  30 

days,  $425.00. 

Credits:   Nov.  18,  $750.00;  Dec.  8,  30-day  note, 

with  interest  at  63^%,  $500.00;  Dec.  29,  $200.00. 

Find  the  equated  date  of  each  of  these  accounts: 

15:    6%,  bankers'  time: 

Debits:  May  17,  $350.00;  May  29,  $255.38;  June 
19,  $125.66;  July  5,  $264.87. 

16:    6%,  exact  time: 

Credits:  Aug.  15,  $325.00;  Aug.  19,  $2,703.59; 
Sept.  14,  $665.32;  Oct.  1,  $225.75. 


24  MATHEMATICS  FOR  THE  ACCOUNTANT 

17:   5%,  exact  time: 

Debits:   June  25,  30  days,  $1,600.00;  July  2,  45 

days,  $800.00;  Aug.  14,  cash,  $500.00. 
Credits:  July  1,  60  days,  $1,200.00;  July  5,  cash, 

$1,000.00. 

18:    5}4%,  exact  time: 

Debits:    Aug.  14,  30  days,  $300.00;  Aug.  22,  45 

days,  $500.00;  Sept.  17,  30  days,  $225.83;  Sept. 

29,  30  days,  $246.73. 
Credits:  Aug.  20,  30  days,  $600.00;  Sept.  19,  cash, 

$250.00. 


CHAPTER  IV 
FOREIGN  EXCHANGE 

The  treatment  of  foreign  exchange  in  this  chapter  is 
wholly  from  the  standpoint  of  arithmetic.  The  financial 
and  economic  phases  of  the  subject  are  fully  treated  in 
many  books  on  the  subject  as  well  as  incidentally  in  most 
books  on  banking  and  foreign  trade. 

WTien  a  sum  of  money  is  to  be  sent  from  one  country  to 
another,  it  is  necessary  to  change  the  value  from  the 
currency  of  the  sender's  country  to  that  of  the  recipient. 
This  process  is  the  arithmetic  of  foreign  exchange,  and 
the  ratio  used  in  making  the  change  is  called  the  rate  of 
exchange. 

In  all  the  great  commercial  nations  the  monetary  sys- 
tem for  international  trade  is  based  on  a  gold  standard.  So 
theoretically  the  rate  of  exchange  between  any  two  coun- 
tries ought  to  be  the  ratio  between  the  amounts  of  gold 
in  their  standard  monetary  units.  This  theoretical  ratio 
is  called  par  of  exchange.  For  instance,  a  United  States 
dollar  weighs  25.8  grains,  and  is  .9  fine;  it  therefore  con- 
tains 23.22  grains  of  pure  gold.  The  English  pound  sterling 
is  113.0016  grains  of  pure  gold.  Par  of  exchange  between 
the  two  countries  is  therefore  the  quotient  of  these  num- 
bers, which  is  4.8665.  This  par  of  exchange  practically 
never  is  the  actual  rate.  For  the  last  few  years  the  rate 
has  fluctuated  very  rapidly  and  violently,  depending  on  the 
ravages  of  the  submarines,  the  varying  fortunes  of  the 
battlefields  all  over  the  earth,  the  purchases  made  in  this 
country  by  England,  and  many  other  circumstances. 

25 


26  MATHEMATICS  FOR  THE  ACCOUNTANT 

The  following  table  gives  par  of  exchange  between 
the  United  States  and  the  other  more  important  com- 
mercial nations: 


Monet  ar}^ 

Value  in 

Country- 

Unit 

U.  S.  Money 

Great  Britain 

pound  sterling 

$4.8665 

Germany 

mark 

.2380 

France 

franc 

.1930 

Belgium 

franc 

.1930 

Switzerland 

franc 

.1930 

Italy 

lira 

.1930 

Greece 

drachma 

.1930 

Spain 

peseta 

.1930 

Russia 

ruble 

.  5150 

Japan 

yen 

.4980 

Holland 

guilder 

.4000 

This  theoretical  par  of  exchange  is  almost  never  the 
actual  rate,  as  was  said  on  the  previous  page.  The  actual 
rate  is  dependent  on  economic  conditions,  such  as  the 
balance  of  trade,  the  supply  of  gold  and  its  location,  labor 
or  other  disturbances,  and  so  on.  As  these  vary  from 
day  to  day,  and  from  hour  to  hour,  so  the  exchange  rate 
varies. 

Rates  of  exchange  are  also  dependent  on  the  kind  of 
paper  to  be  bought  or  sold.  That  is,  on  the  time  between 
the  transaction  and  the  day  when  the  cash  can  actually  be 
collected.  The  usual  kinds  of  paper  are  cables,  demand, 
and  time:  30,  60,  90  or  120-day  paper.  Naturally,  the 
longer  the  time  of  the  paper,  the  lower  the  rate;  just  as 
in  any  discount  transaction.  In  the  case  of  a  time  draft 
on  London,  there  are  also  three  days  of  grace  to  be  con- 
sidered ;  and  the  discount  or  interest  is  reckoned  on  these 
days. 


FOREIGN  EXCHANGE  27 

In  addition  to  the  actual  time  of  the  paper,  bankers 
figure  "transit."  For  instance,  a  60-days'  sight  draft  on 
London  made  on  September  15,  may  not  be  presented  in 
London  until  early  in  October.  It  may  be  held  in  this 
country  several  days  waiting  for  a  steamer;  the  voyage  to 
England  occupies  several  more  days;  and  there  may  be 
some  delay  in  securing  acceptance.  During  all  this  time 
money  is  tied  up  in  the  paper,  and  the  banker  who  sold  it 
is  entitled  to  his  interest.  In  this  chapter,  however, 
transit  will  be  ignored. 

England  has  always  been  the  leader  in  the  world's  trade, 
and  has  controlled  the  gold  market.  So  London  has  be- 
come the  center  of  the  world  of  exchange,  and  sterling 
paper  has  come  to  be  an  acceptable  medium  of  exchange 
in  most  civilized  countries.  This  means  that  the  great 
bulk  of  international  settlements  are  effected  in  this  way. 
So  we  will  study  London  exchange  first,  and  the  applica- 
tion of  the  principles  demonstrated  to  other  monetary 
systems  will  be  obvious. 

The  rate  of  exchange  on  London  is  stated  in  this  country 
as  the  dollars  and  cents  value  of  the  pound  sterling.  It 
fluctuates  by  .0005  or  fractions  or  multiples  thereof.  The 
smaller  English  coins  are  the  shilling,  1/20  or  .05  of  a 
pound;  and  the  penny,  1/12  of  a  shilling,  which  is  usually 
regarded  as  two  cents.  The  penny  is  ordinarily  indicated 
by  the  letter  d.  Thus,  £35  7s.  4d.  means  35  pounds, 
7  shillings,  4  pence. 

Problem:  Change  £43  Us.  7d.  to  U.S.  money, exchange 
at  $4,855: 


£43  Us.  =  £43.55 

43.55  X  4.855 

=  $211.44 

7  X  .02 

.14 

$211.58 


28  MATHEMATICS  FOR  THE  ACCOUNTANT 

Problem:  Change  $368.74  to  English  money,  exchange 
at  $4,878: 

Dividing  368.74  by  4.878  gives  £75.592 
.592  X  20  =  11.84s. 

.84    X  12  =  lOd. 

Answer:  £75  lis.  lOd. 

The  value  of  a  bill  on  London  depends  more  or  less 
directly  on: 

(a)  The  current  rate  of  demand  exchange; 

(b)  The  cost  of  revenue  stamps  to  be  affixed  in  London ; 

(c)  Interest  on  the  money  tied  up  in  the  bill; 

(d)  Commission  of  the  London  correspondent. 
Problem:  What  can  a  banker  afford  to  pay  for  a  bill  on 

London  for  £355  16s.  3d.,  at  60  days'  sight,  demand  rate 
4.87 H,  interest  4%,  stamp  1/20%,  commission  1/4%? 
One  method  of  solution  is  to  reckon  the  price  per  unit 
of  currency,  in  this  case  per  pound  sterling: 

Quotation .A.  875 

From  which  must  be  deducted 

Interest  for  63  days,  exact  time 03366 

Stamp 00244 

Commission 01219     .04829 

He  can  afford  to  buy  the  bill  at 4 .  82671 

times  355.8  and  adding  .06 $1717.3434 

Exporting  gold  is  sometimes  necessary ;  the  gold  may  be 
either  bars  or  coin.  Such  an  occasion  occurs  whenever 
the  exchange  market  is  so  deranged  that  rates  are  exces- 
sively high;  that  is,  whenever  the  balance  of  trade  is 
seriously  disturbed. 

Problem:  Find  the  market  quotation  on  demand  ex- 
change at  which  it  is  advisable  to  export  gold  bars  to  Lon- 
don if  London  pays  77s.  lO^^d.  per  ounce  for  gold  11/12 
fine  (the  English  standard).  The  U.  S.  Treasury  values 
gold  .995  fine  at  $20.67183  per  ounce. 


FOREIGN  EXCHANGE  29 

First  change  77s.  lO^d.  to  pence =  934Md. 

This  is  at  11/12  fine;  now  convert  to 
the  U.  S.  standard,  which  is  prac- 
tically 100%,  by  multiplying  by 
12/11 =  1019.72727d. 

And  1019.72727    -^  240 =      £4.248863 

the  sterUng  value  of  one  ounce  of 
pure  gold. 

Then  20.67183   ^  4.248863 =        4.865263 

the  dollar  value  of  a  pound  sterling 
under  those  conditions. 

To  this  add  charges: 

Freight,  about  1/8% 006082 

Insurance,  1/20% 0024326 

Other  charges,  1/20% 0024326 

Interest,  6%,  20  days 016217 

U.  S.  bar  charge,  40  cents  per 

$1,000 .001946        .0291102 

Cost  per  £  sterling 4.8943732 

That  is,  if  demand  exchange  is  selling  at  more  than 
4.894^/8  it  will  probably  be  cheaper  to  pay  London  in 
gold  bars  rather  than  in  sterling  drafts. 

If  gold  coin  is  exported  the  problem  is  much  the  same 
except  that  there  is  no  bar  charge,  but  an  allowance  of 
about  1/10%  must  be  made  for  abrasion.  We  must  also 
keep  in  mind  that  the  United  States  gold  dollar  is  .900 
fine. 

The  rate  of  exchange  on  France,  Belgium,  Switzerland, 
and  Italy  is  quoted  by  giving  the  exchange  value  of  one 
dollar  in  francs  or  lire.  Thus,  when  exchange  on  Paris  is 
quoted  at  5.18,  a  dollar  will  buy  5.18  francs,  or  5  francs 
18  centimes.  Exchange  on  Paris  fluctuates  by  5/8  of 
a  centime,  about  1  /8  of  a  cent.  These  rates  are  sometimes 
further  modified  by  adding  or  subtracting  1/32%  or  even 


30  MATHEMATICS  FOR  THE  ACCOUNTANT 

1/64%.  Note  that  the  greater  the  numerical  value  of 
the  quotation,  the  lower  the  value  of  French  currency. 
A  market  change  from  5.20  to  5.19^  is  a  rise;  from  5.20 
to  5.205^  is  a  fall. 

The  processes  for  changing  from  French  to  American 
currency  and  vice  versa  are  the  exact  opposites  of  those 
demonstrated  for  English  exchange. 

Problem:  What  is  the  cost  of  a  bill  on  Paris  for  f  1,500 
exchange  at  5.97^4? 

1,500  divided  by  5.9775 =       $250.94 

Problem :  What  number  of  francs  can  be  purchased  for 
$250.00  at  the  same  quotation? 

250  multiplied  by  5.9775 =  f  1,494.375 

The  rate  of  exchange  on  Germany  is  quoted  bj  giving 
the  value  in  cents  of  four  marks.  For  instance,  a  quotation 
on  Berlin  of  .96^  means  that  each  mark  is  worth  24)^ 
cents.  The  following  problems  will  illustrate  the  pro- 
cesses: 

Problem:  How  many  marks  can  be  purchased  for 
$500.00,  exchange  at  73 1^? 

73  J4  divided  by  4 =        .183125 

500  divided  by  .183125 =  M2,730.38 

Problem:  What  is  the  proceeds  of  a  German  bill  for 
M  1,250,  at  the  same  quotation? 

1,250  X  .183125 =  $228.91 

South  American  exchange  has  always  been  uncertain, 
and  now  that  it  is  coming  to  be  based  more  on  the  United 
States  gold  dollar  no  separate  treatment  is  necessary. 

Arbitrage  of  Exchange  is  the  process  of  making  remit- 
tances to  a  country  by  way  of  a  third  country  where  rates 
are  more  favorable.  Exchange  rates  are  now  so  uniform 
that  arbitrage  is  useful  only  rarely. 


FOREIGN  EXCHANGE  31 

Suppose  a  banker  wishes  to  remit  f25,250  to  Paris,  ex- 
change at  5.17 1^.  The  cost  of  the  draft  is  then  $4,879.23. 
But  if  steHing  drafts  cost  4.84,  and  London  quotes  francs 
at  25.25  per  £  sterling,  the  cost  via  London  is  $4,840.00. 
This  is  a  saving  of  $39.23. 

PROBLEMS  ON  CHAPTER  IV 

19.  A  Boston  importer  buys  goods  from  a  Dresden  manu- 
facturer to  the  value  of  M2 1,320.  Find  the  cost  of  a 
bill  of  exchange  at  95^^. 

20.  A  bill  on  London  for  £342  12s.  6d.  is  offered  at  4.86 1^. 
What  is  the  value  in  dollars? 

21.  A  bill  on  Paris  for  f33,250  cost  $6,412.72.  What  was 
the  rate  of  exchange? 

22.  A  Liverpool  merchant  draws  on  an  American  importer 
for  £540 10s.  6d.  What  is  the  value,  exchange  at  4.85  3^? 

23.  What  are  the  proceeds  of  a  documentary  bill  of  ex- 
change on  London  for  £528  8s.  6d.  at  60  days' sight, 
demand  exchange  at  4.77  3^^,  interest  4%,  stamp  and 
commission  as  usual.    Allow  15  days  for  transit. 

24.  Tiffany  and  Co.  import  an  invoice  of  statuettes  from 
Florence  amounting  to  14,725.35  lire.  They  buy  a 
3-days'  sight  draft  on  a  Florence  banker  at  7.3514, 
discount  3%,  stamp  1/20%,  commission  1/8%.  Find 
the  cost  of  the  draft. 

25.  J.  P.  Morgan  and  Co.  remit  to  London  via  Paris  when 
the  rates  are:  New  York  on  London,  4.8675;  New 
York  on  Paris,  5.19^;  Paris  on  London,  25.04.  What 
is  the  gain  on  a  remittance  of  £50,000.00? 

26.  An  importer  can  purchase  certain  goods  in  Amsterdam 
for  15,000  guilders,  exchange  at  38^.  He  can  pur- 
chase the  same  goods  in  Paris  for  f35,000,  exchange  at 
5.85.  Which  is  cheaper,  and  how  much  in  United 
States  currency? 


32  MATHEMATICS  FOR  THE  ACCOUNTANT 

27.  Find  the  cost  of  exporting  enough  gold  to  settle  a 
debt  of  £20,000  if  gold  bars  are  shipped  to  London 
at  77s.  93^d.  per  ounce.  Charges  as  on  preceding 
page. 

28.  A  banking  concern  dealing  in  foreign  exchange  has 
the  following  transactions  on  its  account  with  its 
London  correspondent: 

Debits: 

Sept.    1,  Remittance,  30-day  bill,    £400         @4.86 
10,  Remittance,  sight  bill,       £200  10s.  @  4.87 
15,  Remittance,  demand  bill,  £200  6d.   @  4.8675 
Credits : 

Sept.    2,  Draft,  sight, £300  @  4.87  ^ 

12,  Draft,  demand, £200  12s.  5d.  @  4.87 

20,  Cable,  demand, £100  @  4.88 

Ascertain  the  profit  or  loss  on  the  account  for  the 
month  of  September,  and  state  the  balance  as  of 
September  30,  in  both  dollars  and  sterling,  the  current 
rate  on  that  date  being  4.89.    (Mass.  C.  P.  A.,  1914.) 


CHAPTER  V 
POWERS  AND  ROOTS:   LOGARITHMS 

If  a  number  is  to  be  multiplied  by  itself  a  certain  number 
of  times,  it  is  possible  to  indicate  the  operation  in  very 
brief  form.  For  instance,  if  it  is  necessary  to  multiply 
5  together  four  times,  the  operation  may  be  indicated 
thus:  5^  This  expression  is  then  read,  "5  to  the  fourth 
power."  5X5X5X5  is  625;  and  625  is  called  the 
fourth  power  of  5.  In  general,  5  is  called  the  base,  and 
4  is  called  the  exponent,  showing  how  many  times  5  is 
to  be  multiplied  together. 

Certain  operations  of  compound  interest  work  can  be 
very  much  shortened  by  the  use  of  four  elementary  laws  of 
exponents.    Consider  the  powers  of  10. 


10  X  1                  =  101  = 

10 

10  X  10                 =  10*  = 

100 

10  X  10  X  10             =  10»  = 

1,000 

10  X  10  X  10  X  10         =  10^  = 

10,000 

10  X  10  X  10  X  10  X  10     =  10^  = 

100,000 

10  X  10  X  10  X  10  X  10  X  10  =  10«  = 

1,000,000 

We  can  now  demonstrate  the  four  laws  of  exponents. 
Multiplication:    10»  x  10*  =   100  X   1,000   =    100,000, 
which  is  10^ 

I.  Rule:  To  multiply  powers  of  the  same  base,  add  the 
exponents  and  keep  the  base  unchanged. 

Division:  10«  ^  10^  =  1,000,000  ^  100  =10,000,  which 
is  10^ 

II.  Rule:  To  find  the  quotient  of  powers  of  the  same 
base,  subtract  the  exponent  of  the  divisor  from  the  ex- 
ponent of  the  dividend  and  leave  the  base  unchanged. 

33 


34  MATHEMATICS  FOR  THE  ACCOUNTANT 

Raising  to  a  power:  (lO^)^  =  100  X  100  X  100  = 
1,000,000,  which  is  10^  Evidently  the  exponent  of  the 
result  could  be  obtained  by  multiplying  2X3. 

III.  Rule :  To  raise  a  base  having  an  exponent  to  a  given 
power,  multiply  the  exponents  and  leave  the  base  un- 
changed. 

Finding  a  root:  The  reverse  of  raising  to  a  power  is 
called  finding  a  root;  instead  of  asking  what  is  the  result  if 
we  multiply  100  together  three  times,  we  ask  what  was  the 
number  which  was  multiplied  together  three  times  so  that 
the  result  was  1,000,000 — in  other  words,  what  is  the 
third  root  of  1,000,000.  This  question  is  indicated 
by  the  sign  v^l, 000,000.  The  sign  is  called  a  radical 
sign,  and  the  3  is  called  an  index.  The  answer  to  our 
question  is  evidently  100,  or  10^.  Since  1,000,000  is  10«, 
the  result  could  have  been  obtained  by  dividing  6  by  3. 

IV.  Rule:  To  find  a  root  of  a  base  which  has  an  ex- 
ponent, divide  the  exponent  by  the  index  and  leave  the 
base  unchanged. 

Exponents  may  be  fractional,  negative,  or  zero. 

Fractional  exponents;  It  is  evident  that  in  most  cases 
where  a  root  is  to  be  found,  the  result  will  have  a  fractional 
exponent.  If  we  have  to  find  the  5th  root  of  the  7th 
power  the  result  will  have  the  exponent  7/5.  In  general, 
the  numerator  of  a  fractional  exponent  indicates  a  power 
and  the  denominator  indicates  a  root.  Most  of  the  ex- 
ponents we  deal  with  are  fractional,  and  for  convenience 
the  fractions  are  reduced  to  decimals. 

Negative  exponents :  It  is  also  evident  that  in  division 
the  exponent  of  the  divisor  might  be  larger  than  the  ex- 
ponent of  the  dividend.  This  would  result  in  a  negative 
exponent.  10*  -^  10^  is  a  good  illustration.  10,000  -^ 
1,000,000  =  1/100.    Subtracting  exponents,  we  have  as 


POWERS  AND  ROOTS:  LOGARITHMS  35 

the  exponent  of  the  result  — 2.  So  lO"^  must  mean  the 
same  thing  as  1/100.  In  general,  a  negative  exponent 
indicates  that  the  base  has  become  the  denominator  of  a 
fraction  whose  numerator  is  1.  This  idea  is  important  in 
computing  present  worth. 

The  zero  exponent:  10^  ^  10^  =  10°  because  2—2  =  0. 
Since  100  -^  100  =  1  it  is  evident  that  when  the  exponent 
is  0  the  base  disappears  and  the  value  is  1. 

Logarithms  are  an  ingenious  device  for  using  the  rules 
of  exponents.  If  all  the  numbers,  both  whole,  mixed  and 
fractional,  can  be  expressed  as  powers  of  one  and  the  same 
base,  evidently  these  rules  can  be  applied  in  the  use  of 
such  exponents.  Since  our  number  system  is  decimal,  it 
is  natural  to  select  10  as  the  base  of  this  system  of  logar- 
ithms. Only  a  few  of  these  exponents  are  whole  numbers, 
because  only  a  few  numbers  are  exact  powers  of  10.  The 
following  are  a  few: 


106 

= 

1,000,000 

nl 

6 

105 

= 

100,000 

nl 

5 

10^ 

= 

10,000 

nl 

4 

103 

= 

1,000 

nl 

3 

102 

= 

100 

nl 

2 

101 

= 

10 

nl 

1 

10° 

= 

1 

nl 

0 

10-1 

= 

.1 

nl 

-1 

10-2 

= 

.01 

nl 

-2 

and  so  on.  The  sign  nl  is  used  to  separate  a  number  from 
its  logarithm,  and  is  read  'Hhe  number  whose  logarithm 
is".  The  opposite  sign  hi  ''the  logarithm  of  the  number" 
is  used  in  changing  from  logarithm  to  number,  or  anti- 
logarithm,  as  it  is  called. 


36  MATHEMATICS  FOR  THE  ACCOUNTANT 

In  the  table  of  powers  of  10  note  that  the  logarithm  of 
each  of  the  positive  powers  is  one  less  than  the  number  of 
figures  at  the  left  of  the  decimal  point,  while  in  the  nega- 
tive powers  the  numerical  value  of  the  logarithm  is  one 
more  than  the  number  of  zeros  between  the  point  and  the 
first  significant  digit  (a  significant  digit  is  any  of  the  digits 
from  1  to  9). 

So  far  we  have  spoken  only  of  integral  powers:  powers 
whose  exponents,  or  logarithms,  are  whole  numbers. 
But  between  10  and  100,  for  example,  there  are  89  whole 
numbers  and  an  infinite  number  of  mixed  numbers. 
These  must  all  be  expressed  as  powers  of  10,  and  all  these 
powers  must  lie  between  1,  the  exponent  of  10,  and  2, 
the  exponent  of  100.  Evidently  these  exponents  must 
all  be  1  +  a  decimal.  The  logarithm  of  84,  for  example, 
is  composed  of  two  parts;  a  whole  number  1,  to  indicate 
that  it  is  more  than  10,  and  a  decimal  to  indicate  the 
part  of  the  interval  from  10  to  100.  The  whole  number, 
which  indicates  the  power  of  10  just  below,  is  called  the 
characteristic;  the  decimal  is  called  the  mantissa,  or 
"modifier".  Tables  of  logarithms  never  give  character- 
istics, because  they  can  be  determined  by  inspection, 
keeping  in  mind  the  table  of  powers  of  10. 

Rule  for  determining  the  characteristic: 

If  the  number  has  digits  to  the  left  of  the  decimal  point, 
the  characteristic  is  1  less  than  the  number  of  such 
digits. 

If  the  number  is  a  decimal  fraction,  with  no  digits  to 
the  left  of  the  point,  the  characteristic  is  negative,  and 
is  1  more  than  the  number  of  zeros  between  the  point 
and  the  first  significant  digit. 

For  instance,  the  characteristic  of  317.8895432  is  2.  The 
characteristic  of  .000895  is  — 4.  The  characteristic  of 
7.65  is  0.    This  last  perplexes  some  students,  and  it  is 


POWERS  AND  ROOTS:  LOGARITHMS  37 

important  that  it  be  thoroughly  understood ;  it  is  a  direct 
appHcation  of  the  first  part  of  the  rule. 

Logarithmic  tables,  then,  are  wholly  tables  of  mantissae, 
each  one  corresponding  to  a  certain  series  of  digits. 
There  are  no  decimal  points,  because  the  characteristic 
enables  us  to  locate  them.  It  follows  that  any  series 
of  digits  always  has  the  same  mantissa.  In  the  tables, 
for  instance,  the  mantissa  corresponding  to  8427  is 
925673.    That  is 


S427 

nl 

3.925673 

842.7 

nl 

2.925673 

84.27 

nl 

1.925673 

8.427 

nl 

0.925673 

.8427 

nl 

1.925673 

and  so  on.  This  last  logarithm  is  also  written  1.925673, 
and  9.925673—10.  This  last  method  will  be  followed  in 
this  book,  as  it  simplifies  certain  processes  which  are 
quite  common  in  annuity  work.  The  logarithm  of 
.000895  on  the  preceding  page  would  be  written 
6.951823—10. 

As  was  stated  in  the  first  chapter,  logarithms  should  run 
to  at  least  six  places,  and  seven  are  better.  The  first 
two  digits  of  the  mantissa  change  slowly,  and  are  indi- 
cated only  when  they  change.  This  is  an  important  fact 
to  remember  in  using  tables,  because  the  first  two  digits 
are  apt  to  be  several  lines  above  the  others,  and  must  be 
found  and  used  in  order  to  get  correct  results. 

Finding  a  logarithm  in  the  tables  is  a  simple  matter. 
The  digits  of  the  antilogarithm  are  in  a  column  at  the 
left  of  the  page  and  across  the  top.  To  find  the  logarithm 
or  rather  the  mantissa  of  357.4  turn  to  the  page  or  pages 
containing  the  3,000  series,  and  follow  down  the  left 
margin  until  357  is  reached.     Across  the  top  are  the 


38  MATHEMATICS  FOR  THE  ACCOUNTANT 

digits  from  0  to  9,  and  Id  the  4  column,  opposite  357  we 
read  3,155.  Remembering  that  the  first  two  digits  must 
be  taken  also,  we  read  just  above  55.  The  mantissa  is 
then  553,155.  Now  determine  the  characteristic,  which 
in  this  case  is  2,  and  the  entire  logarithm  is  2.553155. 
If  there  are  less  than  four  digits  in  the  antilogarithm, 
affix  zeros.  For  the  mantissa  of  37  look  for  3,700;  the 
logarithm  is  1.568202. 

Usually  the  problem  is  to  find  the  mantissa  of  a  number 
having  more  than  four  digits.  This  is  known  as  inter- 
polation. To  find  the  logarithm  of  231 .24.  The  character- 
istic is  2.  The  mantissa  is  between  the  mantissae  of 
2,312  and  2,313,  and  evidently  is  4/10  of  the  interval. 

2,313  nl  364,176 
2,312  nl  363,988 
difference  188 

4/10  of  188  is  75.2,  which,  added  to  363,988,  gives 
3,640,632.  Prefixing  the  characteristic  we  have  231.24 
nl  2.3640632. 

The  cologarithm  of  a  number  is  the  logarithm  of  its 
reciprocal.  We  met  a  leciprocal  is  discussing  negative 
exponents,  and  remarked  in  passing  that  the  com- 
monest occurrence  is  in  finding  present  worths.  A  recip- 
rocal is  1  divided  by  a  number.  Now  the  logarithm  of  1 
is  0,  as  was  explained  on  page  21.  To  find  the  cologarithm 
of  387.2,  or  the  logarithm  of  1  divided  by  387.2: 

1  nl         10.000000—10 

387.2         nl  2.587935 

Subtracting  7.412065—10 

One  other  very  common  use  for  cologarithms  is  in  exam- 
ples where  multiplication  and  division  must  be  performed 


POWERS  AND  ROOTS:  LOGARITHMS  39 

together.  The  division  can  be  performed  by  adding  the 
cologarithm  of  the  divisor,  as  will  be  demonstrated 
presently. 

Finding  the  antilogarithm  is  somewhat  more  difficult, 
because  it  almost  always  requires  interpolation.  To  find 
the  number  whose  logarithm  is  2.7851.  The  characteristic 
2  indicates  that  there  are  three  digits  at  the  left  of  the 
decimal  point.  Looking  in  the  tables  for  mantissae 
nearest  7,851  we  find: 

785,116        In        6,097 

7851  the  given  mantissa 

785,045         In         6,096 

The  difference  between  the  logarithms  in  the  table  is 
71;  the  given  logarithm  is  55  more  than  785,045.  The 
required  antilogarithm  is  therefore  55/71  of  the  interval 
from  6,096  to  6,097.  55/71  of  1  is  .774;  affixing  this  to 
6,096  we  have  6,096,774,  and  placing  the  decimal  point 
after  the  third  digit,  we  have  the  antilogarithm  609.6774. 
The  following  problems  will  illustrate  all  the  logarithmic 
processes  which  are  necessary  for  this  book. 

Multiplication:  387.2  X  .04752: 

387.2  nl  2.587935 

.04752  nl  8.676876—10 

Adding  11.264811-10 

or  1.264811 

In  18.3997 

It  is  very  commonly  necessary  to  add  10  — 10  or  multiples 
of  it,  or  to  subtract  them.  Since  10—10  is  0  we  evidently 
do  not  change  the  value  of  any  expression  by  this  device. 
The  addition  of  the  logarithms  is  in  accordance  with  the 
first  law  of  exponents,  in  multiplication,  page  33. 


40 


MATHEMATICS  FOR  THE  ACCOUNTANT 


Division;  69.45  -^  3.894 
69.45 
3.894 
Subtracting 
In 


nl 
nl 


1.841672 
.590396 


1.251276 
17.835123 

The  subtraction  is  in  accordance  with  the  second  law  of 
exponents,  in  division. 
Multiplication  and  division  in  the  same  example: 
68.39  X  31,750 


243.6 

68.39 

nl 

1.834993 

31750 

nl 

4.501744 

243.6 

n — colog 

7.613323—10 

Adding 

13.950060—10 

or 

3.950060 

In 

8913.7347 

Raising  to  a  power: 

a)     (2.798)'* 

2.798 

nl 

.446848 

times  8 

3.574784 

In 

3756.504 

b)    to  illustrate  treatment  of  a  negat 

ive  characteristic 

(.0693)5 

.0693 

nl 

8.840733—10 

times  5 

44.203665—50 

or 

4.203665—10 

In 

.0000015983 

Multiply  by  the  exponent;  see  rule  III 

- 

Finding  a  root: 

a)     W968.4 

968.4 

nl 

2.980055 

divided  by  7 

.4265793 

In 

2.67042 

POWERS  AND  ROOTS:  LOGARITHMS 


41 


b)    to  illustrate  treatment  of  a  negative  characteristic 


>J/.08355 

.08355  nl 

after  we  divide  by  4  the  charac- 
teristic must  be  — 10;  so  before 
division  it  must  be  — 40 ;  so  it  is 
necessary  to  add  30 — 30,  mak- 
ing the  log 
now  divide  by  4 
In 


8.921946-10 


38.921946^0 
9.7304865-10 
.537634 


Divide  by  the  index  as  in  rule  IV,  page  34. 


Combinations  of  these  processes  are  rarely  necessary, 
a)    (3.066)11 


^12.74 

3.066              nl 

.486572 

times  11 

5.352292 

12.74                 nl 

1.105169 

divided  by  5 

.2210338 

Subtracting 

5.1312582 

In 

153287.6 

4.925  X    >J/16740 

(38.25)^ 

4.924                nl 

.692406 

16740                nl 

4.223755 

divided  by  3 

1.407918 

38.25                 nl 

1.582631 

times  4 

6.339524 

divide  by  adding  ( 

jolog 

3.669476—10 

Adding 

5.769800-10 

hi 

.0000588573 

42  MATHEMATICS  FOR  THE  ACCOUNTANT 

PROBLEMS  ON  CHAPTER  V 

A.  Problems  to  give  facility  in  use  of  tables: 

29.  Find  the  logs  of  432.5        4.325         .004325 

30.  Find  the  logs  of  5700  5.7         570000 

31.  Find  the  logs  of  78.9342        05.4386        7762.47 

32.  Find  the  cologs  of  13.97        .04953        620000 

33.  Find  the  cologs  of  6.91843        100.764        .000015 

34.  Find  the  antilogs  of  3.270679  1.284656  8.817962—10 

35.  Find  the  antilogs  of  1.316274  2.967777  4.830034—10 

36.  Find  the  antilogs  of  .00317255 

6.779999—10  3.84506234 

B.  Perform  the  following  by  the  use  of  logs: 

37.  Multiplication: 

378.2  X  56.43 

.06925  X  34700 

616500  X  3.9483  X  .00745 

38. 


39. 


Division : 

930.07   - 

-  1.05 

.58023   - 

-  7.7438 

58.023   - 

-  .077438 

8.9094   - 

-  .89094 

Multiplicatic 

>n  and  division: 

9.1932  > 

:  .5307 

.0076 
6800000  X  .0038902 


1.025 

40.  Raising  to  a  power; 

(3.4684) « 
(1.01225)14 

41.  Raising  to  a  power: 

(.6307)9 
(.1213)6 


POWERS  AND  ROOTS:  LOGARITHMS  43 


42.  Finding  a  root: 

>^810.5 
^1.0575 

43.  Finding  a  root: 

V.0069183 
^.0003301 

Miscellaneous : 

44.  4.6173  X  (2.837)^ 

.10765  X  48.27 


45.  /3  X  7.5 
y  (1.015)4 

46.  Ay392.7  X  61.3455  X  33.9 

(1.0175)«  ~ 

47.  r2.94635  X  ^671  e 


L  1.023 

48.  ^300.75  X  .000045 

(.8512)8 


CHAPTER  VI 
COMPOUND  INTEREST  AND  PRESENT  WORTH 

At  this  point  we  enter  on  the  study  of  what  the  Ameri- 
can Institute  of  Accountants  calls  Elementary  Actuarial 
Science.  An  actuary  is  the  mathematical  expert  of  a  life 
insurance  company.  It  is  his  business  to  calculate  the 
income  and  expenditures  of  the  company  and  see  that  the 
company  is  on  a  safe  financial  basis:  to  see  that  it  has 
sufficient  funds  on  hand  to  meet  all  ordinary  calls  and  an 
additional  amount  to  meet  emergencies.  These  expendi- 
tures are  concerned  mostly  with  payments  on  account  of 
policies,  either  life  or  endowment.  Such  expenditures 
therefore  can  be  estimated  pretty  closely  if  there  are 
sufficient  data  regarding  the  number  of  persons  who  will 
live  or  die  during  the  year.  In  other  words,  the  actuary  is 
concerned  with  the  probabilities,  as  they  are  called.  By 
the  study  of  many  thousand  lives  actuaries  have  been 
enabled  to  formulate  laws  stating  that  out  of  a  certain 
number  of  persons  alive  on  a  given  date  a  stated  propor- 
tion will  live  through  an  entire  year  and  the  rest  will  die 
within  the  year.  These  contingencies,  so  called,  affecting 
sometimes  a  single  life  and  sometimes  a  group  of  lives, 
make  the  mathematics  of  the  actuary  exceedingly  diffi- 
cult. 

Moreover,  these  calculations  must  be  based  on  a  long 
period  of  years.  The  mortality  tables  for  straight  life 
policies  run  to  the  ninety-sixth  year.  Most  endowment 
insurance  is  for  periods  of  ten  to  twenty-five  years.  Under 
these  circumstances  all  calculations  must  be  on  a  com- 
pound interest  basis.    The  series  of  payments  on  an  en- 

44 


COMPOUND  INTEREST  AND  PRESENT  WORTH  45 

dowment  insurance  policy  are  invested  by  the  company 
and  earn  interest.  It  is  reasonable  that  the  income  on 
the  payments  of  each  policyholder  should  be  credited  to 
his  policy. 

The  mathematics  which  have  been  outlined  in  the  last 
two  paragraphs  are  Actuarial  Science.  The  effect  of 
probability  on  annuities  means  nothing  to  the  accountant, 
however,  because  he  assumes  that  the  financial  arrange- 
ments which  he  supervises  will  be  permanent  in  most 
cases.  To  describe  this  situation,  mathematicians  speak 
of  annuities  certain;  annuities  which  will  not  be  affected 
by  the  death  or  mishap  of  any  person  or  persons,  but  will 
continue  to  the  limit  of  the  time  assigned  them.  This  is 
Elementary  Actuarial  Science. 

A  very  common  way  of  referring  to  compound  interest  is 
"interest  on  interest".  This  definition  is  not  true,  and 
is  misleading.  It  is  misleading  because  it  implies  usury. 
No  such  taint  inheres  in  compound  interest;  every  dollar 
of  capital  is  always  at  work,  and  inasmuch  as  interest 
rates  are  stated  for  definite  periods,  a  year,  six  months, 
three  months,  at  the  end  of  the  period  the  interest  which 
has  been  accruing  becomes  due  and  is  no  longer  interest, 
but  an  addition  to  capital.  So  again  it  is  not  interest  on 
interest,  but  interest  on  a  periodically  increasing  capital. 
The  chief  argument  for  compound  interest  is  this.  If  a 
sum  of  money  is  invested,  the  interest  is  supposed  to 
be  paid  when  due.  It  is  loaned  fcr  the  sake  of  this  in- 
come. If  the  interest  is  not  paid  promptly ,  part  of  the 
income  is  lost  and  the  interest  rate  is  lowered.  But  this 
interest  is  earning  money  for  the  party  with  whom  the 
money  is  invested.  So  he  and  not  the  rightful  owner  is 
reaping  the  benefit  of  the  investment. 

What  is  the  significance  for  the  accountant?  In  the  first 
place,  many  of  the  transactions  of  the  business  world  are 


46  MATHEMATICS  FOR  THE  ACCOUNTANT 

already  on  a  compound  interest  basis.  This  applies  es- 
pecially to  investment  accounting;  to  bond  accounts  and 
businesses,  savings  banks,  insurance  companies,  and  even 
the  interest  on  checking  accounts  in  commercial  banks. 
And  many  accountants  in  supervising  bond  issues  of 
public  and  private  corporations  say  that  they  build  the 
sinking  fund  schedules  on  a  compound  interest  basis 
whenever  the  accounting  staff  of  the  concern  are  capable 
of  handling  schedules  built  in  that  way. 

Evidently,  if  an  accountant  is  to  audit  the  accounts  of 
such  a  concern  he  must  know  his  annuities  thoroughly. 

Again,  as  has  been  said  already,  every  business  has  its 
fiscal  period,  and  at  the  end  of  that  period,  the  net  earnings 
are  transferred  to  capital  in  any  one  of  a  number  of  ways. 
If  the  investment  in  the  business  is  already  adequate,  any 
overplus  of  capital  will  be  invested  in  other  businesses  or 
in  securities,  or  at  least  in  some  banking  house  which 
offers  a  fair  rate  of  interest.  In  some  way  this  money  will 
earn  an  income  at  not  less  than  the  savings  bank  rate. 

Usually  an  additional  investment  in  the  business  is  de- 
sirable, which  may  take  the  form  of  a  direct  increase  in 
surplus,  or  may  be  apportioned  among  the  various  re- 
serves, for  sinking  funds,  depreciation,  or  so  on.  Such 
additional  investments  of  course  earn  an  additional  in- 
come, just  as  the  previous  investment  did.  If  the  business 
man  expected  an  8%  return  on  his  original  investment  he 
will  hope  for  an  8%  return  on  the  increase.  And  it  seems 
reasonable  that  such  income  should  be  credited  to  the 
surplus  or  reserve  on  which  it  was  earned.  Because  if 
such  reserves  were  represented  by  funds  in  the  hands  of 
a  trustee,  they  would  earn  an  income  at  not  less  than  the 
savings  bank  rate,  and  this  income  would  not  be  trans- 
ferred to  the  free  surplus  of  the  company,  but  would  be 
added  to  the  fund  in  the  hands  of  the  trustee  and  re- 
invested by  him. 


COMPOUND  INTEREST  AND  PRESENT  WORTH  47 

These  are  the  arguments  for  compound  interest.  They 
are  the  arguments  from  reason  and  consistency  and  from 
the  best  custom.  Failure  to  recognize  them  causes  mis- 
statement of  income  and  failure  to  allocate  it  properly 
between  the  various  parts  of  free  or  appropriated  capital 
on  which  it  was  earned.  Compound  interest  is  actually 
at  work  whether  realized  or  unrealized;  acknowledgment 
of  it  as  a  fact  is  the  duty  of  the  progressive  accountant. 

Computation  of  compound  amount :  We  have  been  dis- 
cussing compound  interest.  As  a  matter  of  fact  we  usually 
find  the  compound  amount  first,  and  then  deduct  the 
principal,  the  result  being  the  interest.  The  reason  for 
this  is  that  there  is  a  very  simple  process  for  finding  the 
compound  amount  of  $1.00  for  any  time  at  any  rate; 
and  since  all  dollars  in  a  given  transaction  increase  at  the 
same  rate,  the  compound  amount  of  any  principal  is 
merely  the  compound  amount  of  $1.00  multiplied  by  the 
given  principal. 

We  have  perhaps  been  accustomed  to  find  compound 
amount  by  a  laborious  process,  thus: 

To  find  the  compound  amount  of  $1.00  at  5%  for  five 


years : 

1st  year 

$1.00 

add  5% 

.05 

2nd  3'ear,  principal 

1.05 

add  5% 

.0525 

3rd  year,  principal 

1.1025 

add  5% 

.055125 

4th  year,  principal 

1.157625 

add  5% 

.05788125 

5th  year,  principal 

1.21550625 

add  5% 

.06077531 

Compound  amount 

$1.27628156 

48  MATHEMATICS  FOR  THE  ACCOUNTANT 

Of  course  no  savings  bank  or  savings  department  of  a 
commercial  bank  carries  its  interest  to  eight  places  of 
decimals.  But  this  must  be  done  in  investment  mathe- 
matics, where  the  items  which  are  being  accumulated  may 
run  into  milUons  of  dollars. 

A  much  simpler  way  to  solve  the  problem  is  given  by  an 
elementary  principle  of  algebra.  The  principal  is  1 ;  the 
rate  of  interest,  hereafter  called  i,  is  .05.  The  amount 
at  the  end  of  one  period  is,  therefore,  1.05.  We  are 
compounding  annually,  so  the  mmaber  of  conversion  in- 
tervals, as  they  are  called  (the  number  of  times  that  the 
interest  becomes  an  addition  to  principal)  is  5;  this  time 
element  is  indicated  by  n.  Now  if  we  multiply  1.05 
together  5  times,  or,  as  the  mathematicians  say,  raise  it 
to  the  fifth  power,  the  product  mil  be  $1.27628156,  the 
very  same  result  as  we  obtained  by  the  other  method. 
In  general,  to  find  the  compound  amount  of  1  for  n  periods 
at  rate  i,  raise  1  +  i  to  the  n-th  power.  Expressed  as  a 
formula  this  is  (1  +  i)°.  This  is  indicated  conmionly  by 
the  symbol  s"^. 

It  is  important  to  remember  that  n  is  the  number  of 
conversion  intervals,  not  the  number  of  years.  For  in- 
stance, if  a  savings  bank  states  that  its  rate  is  4%,  it 
usually  means  2%  every  six  months.  A  dollar  left  in  this 
bank  five  years  will  amount  to  the  tenth  power  of  1.02. 

For  ordinary  times  and  rates  there  are  good  tables  of 
compound  amount. 

Problem:  To  find  the  compound  amount  of  $150.00  for 
17  years  at  5%  compounding  semiannually.  The  rate 
is  23^%  and  the  time  is  34  periods.  In  the  2^/2%  column 
of  the  tables  opposite  34  of  the  time  column  read, 
2.31532213.  Multiplying  by  150  gives  the  compound 
amount  $347.2983.    The  compound  interest  is  $197.2983. 


COMPOUND  INTEREST  AND  PRESENT  WORTH        49 

For  unusual  rates,  which  often  occur,  logarithms  are 
necessary.  And  they  should  be  at  least  ten-place  logar- 
ithms, as  was  stated  in  the  first  chapter. 

Problem:  Find  the  compound  amount  of  $246.75  for  12 
years  at  2^%  compounding  annually. 
i  is  .02375 
1  +  i  is  1.02375 
n  is  12. 

We  have  to  find  (1.02375)'2 
1.02375        nl         .0101939148 
times  12  .1223269776 

In  1.325338 

tunes  246.75  =  $327,027 
Nominal  and  effective  rates:  At  the  beginning  of  the 
chapter  on  interest  one  element  of  the  interest  problem 
mentioned  was  "frequency",  the  length  of  the  "con- 
version interval".  For  instance,  savings  banks  and  de- 
partments state  a  yearly  rate,  but  compound  semiannually 
or  even  monthly.  Bonds  usually  pay  semiannual  interest, 
and  this  is  considered  when  the  price  is  quoted.  These 
are  only  two  common  instances  out  of  many.  When  a 
bank  advertises  compound  interest  at  4%,  convertible 
semiannually,  it  usually  means  to  pay  2%  every  six 
months;  the  actual  yearly  rate  is  4.04%.  In  this  case 
4%  is  called  the  nominal  rate,  indicated  by  j,  and  4.04% 
is  called  the  effective  rate,  indicated  by  i.  If  the  com- 
pound amount  of  $1.00  for  five  years  at  4%  nominal, 
convertible  semiannually,  is  desired,  i  is  .02,  1  +  i  is  1.02, 
and  T)  is  10.  In  the  table  of  s  °  in  the  2%  column  opposite 
10  read  1.21899442.  Similarly,  the  compound  amount 
of  $1.00  for  15  years  at  6%  nominal,  convertible  quarterly, 
is  Ihe  compound  amount  for  60  periods  at  1^%,  or 
2.44321978.  Such  problems  usually  lead  to  unusual  rates 
and  require  the  use  of  logarithms  in  solution. 


50  MATHEMATICS  FOR  THE  ACCOUNTANT 

Evidently,  given  a  nominal  rate,  the  shorter  the  con- 
\'ersion  interval,  the  higher  the  effective  rate. 

At  6%  compounding  yearly  (1  +  i)  °  is  1.06 

"  semi-annually  1.0609 

«  quarterly  1.061364 

"  monthly  1.061678 

daily  1.061826 

There  is  a  limit  to  this  increase,  however.  For  the  nominal 
rate  6%,  the  highest  possible  effective  rate,  for  instan- 
taneous compounding,  is  slightly  less  than  6.184%.  This 
idea  of  instantaneous  compounding,  or  continuous  com- 
pounding, is  treated  at  length  by  the  books  on  Actuarial 
Science.  The  rate  6.184%  is  called  "force  of  interest". 
It  has  no  significance  for  the  accountant. 

To  change  from  nominal  to  effective  rates: 


[i  +  i] 

L  mJ 


The  formula  is  i  =   1 1  +  - 1    —1 

m- 

where  i  is  the  effective  rate,  j  is  the  nominal  rate,  and  m 
is  the  number  of  conversion  intervals  in  one  year. 

Rule:   Find  the  compound  amount  of  the  periodic  rate 
for  m  periods  and  subtract  1. 

Problem:    What  is  the  effective  rate  when  the  yearly 
nominal  rate  is  3%,  and  compoundings  are  semiannual? 
j  is  .03 

m  is  2 

J-  is  .015 
m 

.1  -h  ^  is  1.015 
m 

Then  the  effective  rate  is  (1.015)2—1,  or  .030225. 
Stated  as  a  percent  it  is  3.0225%. 


COMPOUND  INTEREST  AND  PRESENT  WORTH        ol 

Nominal  rate  when  the  effective  rate  is  given:  Insurance 
companies  and  some  other  concerns  which  have  a  great 
variety  of  investments  having  varying  interest  dates  and 
periods  find  it  advisable  to  annuahze  all  interest  rates. 
This  leads  to  a  situation  which  is  rather  difficult  to  grasp 
at  first.  They  assume  a  yearly  rate  of  income,  say  6%. 
Now  if  a  certain  investment  pays  semiannual  interest  and 
is  to  be  valued  at  6%  annually,  it  is  evident  that  6%  is 
the  effective  rate.  What  is  the  semiannual  rate?  Not 
3%,  for  that  would  yield  an  annual  rate  of  6.09%,  as  was 
stated  on  the  previous  page.  Referring  to  the  discussion 
of  roots  on  page  20,  the  rate  which  after  two  conversions 
yields  1.06  must  be  the  square  root  (or  second  root)  of 
1.06.  If  (1  +  i)2  =  1.06,  then  1  +  i  =  >yi.06.  This 
solution  of  such  a  problem  is  logarithmic  and  based  on 
the  fourth  rule  of  exponents. 

1.06  nl  .0253058653 

dividing  by  2  .01265293265 

In  (10-place  table)  1.029563 

or  2.9563%  per  period 

that  is  5.9126%  is  the  nominal  rate. 

Such  a  nominai  rate  is  indicated  by  the  symbol  j  p,  where  p 
indicates  the  number  of  compoundings  per  year.  A  table 
of  the  values  of  j  p  for  the  usual  rates  and  for  two,  four 
and  twelve  compoundings  yearly  is  given  in  all  good 
collections  of  tables. 

To  illustrate  a  little  differently,  the  value  of  j  4  for  5% 
is  given  in  that  table  as  .0490889.  This  is  a  nominal  rate, 
which  means  that  the  periodic  rate  is  1/4  of  the  annual 
rate.  Now  1/4  of  .0490889  is  .0122722.  Then,  according 
to  the  definition  of  a  periodic  rate  given,  1.0122722  raised 
to  the  fourth  power  ought  to  yield  1.05.  By  actual  multi- 
plication we  find  the  rate  to  be  1.0500025.  The  error  is  of 
course  due  to  rounding  up  whenever  digits  were  discarded. 


52  MATHEMATICS  FOR  THE  ACCOUNTANT 

Formula  for  finding  the  nominal  rate  when  the  effective 
rate  is  given  is 

j  P  =  p  (^1+i  -  1) 

Rule:  Find  the  p-th  root  of  1  +  i  and  subtract  1 ;  this  is 
the  periodic  interest  rate;  multiply  by  p  to  find  the  annual 
nominal  rate. 

To  sum  up,  rates  are  always  given  on  a  yearly  basis.  If 
compoundings  are  other  than  yearly,  there  may  be  two 
ways  of  stating  the  rate.  Either  the  rate  given  is  a  nominal 
rate,  in  which  case  the  periodic  rate  is  the  n-th  part  of  the 
given  rate.  Or  the  rate  may  be  effective,  in  which  case 
the  periodic  rate  is  the  n-th  root  of  the  given  rate. 

Present  worth  and  compound  discount:  If  a  sum  of 
money  is  due  at  some  future  date,  it  can  be  discounted  to 
the  present  time  by  deducting  interest  from  the  present  to 
the  due  date.  This  is  a  common  process  on  short  time 
notes,  in  which  case  the  interest  is  simple  interest  and  the 
process  a  very  simple  one.  But  in  the  case  of  long  time 
transactions,  where  compound  interest  must  be  reckoned, 
the  process  is  somewhat  different.  There  is  an  old  prin- 
ciple of  arithmetic  that  if  the  product  of  two  numbers  is 
known  and  also  one  of  the  factors,  the  other  factor  can 
be  found  by  division.  Since  compound  interest  is  found 
by  multiplication,  evidently  present  worth  on  a  com- 
pound interest  basis  is  found  by  division.  And  since  all 
our  compound  amounts  are  based  on  1,  and  expressed  by 
the  formula  (1  +  i)  °,  evidently  present  worth  of  1  is  1 
divided  by  (1  +  i)  °,  or  expressed  as  a  formula, 

present  worth  of  1  == 


(1  +  i) " 

Rule:   Divide  1  by  the  compound  amount  of  1  for  the 
given  time  and  rate. 


COMPOUND  INTEREST  AND  PRESENT  WORTH        53 

Present  worth  is  indicated  by  the  symbol  v  °.  Tables 
for  the  usual  rates  and  times  are  given  in  all  good  collec- 
tions of  tables. 

Problem:   Find  the  present  worth  of  S125.00  due  in  12 
years  on  a  6%  nominal  basis,  convertible  semiannually. 
1  is  3% 
n  is  24 

In  the  table  of  v  °,  3  %  column,  opposite 
24  read  .49193374 
times  125    $61.4917. 
Problem:   (to  illustrate  use  of  logarithms  when  the  rate 
is  unusual):     Find  the  present  worth  of    1  due  in  two 
years  at  3M%  nominal,  convertible  semiannually, 
i  is  .01625 
n  is  4 

1.01625        nl  .0070005586 

times  4  .0280022344 

colog  9.9719977656—10 

In  .937557 

Note  that  since  the  (1  +  i)  °  is  in  the  denominator,  the 
value  of  V "  always  requires  finding  of  a  cologarithm. 

SUMMARY  OF  THE  PROCESSES  USED  IN  COM- 
POUND INTEREST  COMPUTATION 

Compound  amount:  s  °  =  (1  +  i)  °. 

Problem:  Find  the  compound  amount  of  $1.00  at  4 1^2% 
nominal,  convertible  semiannually,  for  20  years. 

1  =  2M% 

n  =  40 

1.0225     nl      .0096633167 

times  40  .3865332668 

In  2.435189 


54  MATHEMATICS  FOR  THE  ACCOUNTANT 

Present  worth:  v  °  =    — - 

(1  +  i)  " 

Problem:    Find  the  present  worth  of  11.00  due  in  22 
years  at  33^%  nominal,  convertible  semiannually. 


1  =  i'AVo 

n  =  44 

1.0175 

nl 

.0075344179 

times  44 

.3315143876 

colog 
In 

9.6684856124-10 
.466107 

Present  worth  of  an  interest  bearing  debt: 

Given  a  debt  due  in  n  periods  with  interest  at  rate  k  per 

period.    To  discount  it  to  the  present  time  at  rate  i  per 

period. 
/  There  are  two  cases,  one  when  the  conversion  intervals 

at  rates  k  and  i  are  the  same,  and  the  other  case  when 

the  conversion  intervals  differ.    Let  m  be  the  number  of 

periods  in  the  term  of  discount. 

The  formula  is  v  = — 

(l  +  i)- 

Rule:  Find  the  compound  amount  of  1  at  rate  k,  and 
divide  by  the  compound  amount  (or  multiply  by  the 
present  worth)  of  1  at  rate  i. 

Problem:  A  debt  of  1  bears  interest  at  5%  annually,  and 
is  due  in  12  years.  What  is  the  present  worth  at  6% 
annually. 

(1.05)12  =  1.79585633 

-^^ =  .49696936 

(1.06)12 

product  .8924856 

Problem  (to  illustrate  differing  conversion  intervals) : 
Find  the  present  worth  of   $155.63  due  in  5  years  with 


COMPOUND  INTEREST  AND  PRESENT  WORTH 


55 


interest  at  5%  nominal,  convertible  semiannually,  dis- 
counted at  6%  annually: 


(1.0125) 
1 

(1.06)^ 
product 


1.28008454 

.74725817 
.9565536 


times  155.63    =       $148.86843 
To  change  from  nominal  rate  to  effective: 


The  formula  is  i  =  (1  +  — ) 

m 


-1 


Problem :  WTiat  is  the  effective  rate  when  the  stated  rate 
is  6%,  compounded  monthly. 

J       =     6% 


m    = 

± 

m 

1.005 

times  12 

In 

subtract  1 

or 


12 
.005 


nl 


.0021660618 
.0259927416 

1.0616774 
.0616774 

6.16774% 

To  change  from  effective  to  nominal: 


The  formula  is  jp  =  p  (1/1  +  i— 1) 

Problem:  What  is  the  nominal  rate  equivalent  to  an 
effective  rate  of  434%,  i^  there  are  four  conversion  inter- 
vals in  the  year. 


1.0425  nl 

divided  by  4 

In  (10-place  table) 

subtract  1 

or 

times  4 


.0180760636 

.0045190159 
1.0104597 

.0104597 
1.04597%  periodic  rate 
4.18388%  yearly 


56  MATHEMATICS  FOR  THE  ACCOUNTANT 

To  find  the  rate,  given  principal,  amount,  and  time: 
The  formula  is  i  =    „/  -  —I 

Rule:    Divide  the  amount  by  the  principal;  find  n-th 
root  of  the  quotient;  subtract  1. 

Note  that  usually  the  easiest  way  to  divide  S  by  P  is  to 
find  the  difference  of  the  logarithms. 

Problem:    At  what  annual  rate  will  $1.00  amount  to 
$2.00  in  27  years. 

S  divided  by  P         =  2. 

2  nl  V.30103 

divide  by  27  .0111492555 

In  (10-place  tables)  1.026045 

subtract  1  and  the  rate  is   2.6045% 
Problem:    At  what  nominal  rate,  compounding  quar- 
terly, will  $3.00  amount  to  $5.00  in  10  years. 

.69897 
.477121 
.221849 
.005546225 
1.012853 

1.2853%  periodic  rate 
5.1412%  annually. 

To  find  the  time,  given  principal,  amount,  and  rate: 

,    .  log  S— log  P 

The  formula  is  n  = -; — ,    ., 

log  (1  +  i) 

Rule:  From  log  of  S  subtract  log  of  P;  divide  by  log  of 

1  +  i.  -• 

Problem:    In  what  time  will  $2.00  amount  to  $3.00  at 

5%  annually? 

3  nl  .477121 

2  nl  .301030 


5 

nl 

3 

nl 

subtract 

divide  by  40 

In 

or 

times  4 

COMPOUND  INTEREST  AND  PRESENT  WORTH  S7 

subtract  .176091 

1  +i  =  1.05    nl  .021189 

quotient  8.31  years. 

The  quotient  should  not  be  changed  to  months  and  days, 
because  compound  interest  is  not  assumed  for  parts  of 
periods.  In  this  type  of  problem,  and  whenever  the 
logarithm  of  1  +  i  is  used  as  a  divisor,  the  6-place  man- 
tissa is  sufficient;  if  the  10-place  mantissa  were  used  it 
would  be  cumbersome  and  we  would  begin  to  cut  off  digits 
at  once.  Note  also  that  although  one  logarithm  is  divided 
by  another,  the  quotient  is  not  a  logarithm.  This  situa- 
tion always  occurs  in  time-problems. 

When  the  rate  is  nominal,  it  is  necessary  to  proceed  as 
follows : 

Problem:  In  what  time  will  $1.00  amount  to  $2.00  at 
5%  nominal,  convertible  quarterly? 


periodic  rate  i 

is 

.0125 

2 

nl 

.301030 

1 

nl 

.000000 

subtract 

.30103 

1.0125 

nl 

.005395 

quotient 

55.8  periods 

or 

13.95  years 

Compare  this  w 

ith  the  approximation 

commonly  used, 

n  = 

.693 
i 

+  .35 

.693 

in  this  case \-  .35  =  55.79  periods. 

.0125 


58  MATHEMATICS  FOR  THE  ACCOUNTANT 

PROBLEMS  ON  CHAPTER  VI 

A.  Simple  problems  to  be  solved  by  use  of  the  tables  and 
multiplication  or  division : 

49.  What  is  the  compound  amount  of  $1.00  for  15  years 
at  6%  annually? 

50.  What  is  the  compound  amount  of  $1.00  for  15  years 
at  6%  nominal,  convertible  semiannually? 

51.  What  is  the  compound  amount  of  $383.97  for  W}4 
years  at  5%  nominal,  convertible  quarterly? 

52.  What  is  the  compound  amount  of  $1,500.00  for  22 
years  at  6%  nominal,  convertible  quarterly. 
(Note:  If  the  table  does  not  include  values  for  88 
periods,  they  can  be  obtained  by  multiplying  the 
value  for  85  periods  by  the  value  for  3  periods.  See 
the  first  law  of  exponents.) 

53.  What  is  the  compound  amount  of  $2,785.94  for  20 
years  at  3%  nominal,  convertible  semiannually? 

54.  What  is  the  effective  rate  equivalent  to  3%%  nom- 
inal, convertible  quarterly? 

55.  What  is  the  present  worth  of  $1.00  due  in  17  years 
at  6%  annually. 

56.  What  is  the  present  worth  of  $1.00  due  in  17  years 
at  5%  nominal,  convertible  quarterly? 

57.  What  is  the  present  worth  of  $1,934.17  due  in  11 
years  at  6%  nominal,  convertible  semiannually? 

B.  More  difficult  problems,  usually  requiring  the  use  of 
logarithms  in  solution : 

58.  What  is  the  compound  amount  of  $1.00  for  11  years 
at  4:}4%  nominal,  convertible  quarterly? 

59.  What  is  the  compound  amount  of  $1.00  for  16  years 
at  6%  nominal,  convertible  monthly? 

60.  What  is  the  compound  amount  of  $1.00  for  11  years 
at  4^%  nominal,  convertible  semiannually? 


COMPOUND  INTEREST  AND  PRESENT  WORTH  59 

61.  What  is  the  effective  rate  equivalent  to  3^%  nom- 
inal, convertible  monthly? 

62.  What  is  the  effective  rate  equivalent  to  7^%  nom- 
inal, convertible  quarterly? 

63.  What  nominal  rate,  compounded  quarterly,  is  equiv- 
alent to  6%  effective? 

64.  What  nominal  rate,  compounded  monthly,  is  equiv- 
alent to  534%  effective? 

65.  What  is  the  present  worth  of  $1.00  due  in  11  years, 
^t  434%  nominal,  convertible  semiannually? 

66.  What  is  the  present  worth  of  $1.00  due  in  8  years  at 
3.9%  nominal,  convertible  monthly? 

67.  At  what  annual  rate  will  $15,000  double  itself  in 
25  years? 

68.  At  what  nominal  rate,  convertible  quarterly,  will 
$3,000.00  amount  to  $5,000.00  in  5  years? 

69.  The  British  government  sold  non-interest  bearing 
certificates  for  15s.  6d.,  redeemable  after  5  years  at 
£1.  What  is  the  equivalent  savings  bank  rate, 
nominal,  convertible  semiannually. 

70.  Benjamin  Franklin  bequeathed  $5,000.00  to  the  City 
of  Boston.  After  104  years  it  had  amounted  to 
$431,000.00.  What  is  the  equivalent  rate,  nominal, 
convertible  quarterly? 

71.  In  what  time  will  $1.00  double  itself  at  6%  annually? 
Solve  by  logs  and  check  the  solution  by  the  approx- 
imation formula. 

72.  $5,000.00  is  invested  at  5%  nominal,  convertible  semi- 
annually.   In  what  time  will  it  amount  to  $7,500.00? 


CHAPTER  VII 
ANNUITIES:  AMOUNT  AND  PRESENT  WORTH 

An  annuity  is  a  series  of  equal  payments  at  equal 
intervals  of  time,  accumulated  at  a  fixed  rate  of  interest. 
The  amount  of  the  periodic  payment  is  always  stated 
on  a  yearly  basis.  A  good  example  of  this  is  the  in- 
terest on  a  bond.  The  bond  is  advertised  as  a  5%  bond; 
but  the  issuing  or  selling  concern  means  that  the  bond 
will  pay  2y-f/o  every  six  months.  Another  important 
fact  to  remember  is  that  payments  occur  at  the 
end  of  the  vear  or  other  period.  This  is  reasonable,  as 
will  appear  if  we  consider  the  source  of  the  money  with 
which  the  payments  are  made.  A  corporation  pays  the 
interest  on  its  bonds  out  of  income.  This  income  is 
reckoned  at  the  end  of  the  fiscal  period.  A  person  paying 
interest  on  a  mortgage  does  so  at  the  end  of  the  interest 
period.  There  are  numerous  other  instances  which  will 
occur  to  any  accountant.  The  most  common  instances 
of  payments  at  the  beginning  of  the  period  are  premiums 
on  insurance  policies  and  some  rental  payments.  Such  an 
annuity  is  called  an  annuity  due  and  calls  for  special 
treatment.  There  is  also  the  deferred  annuity,  in  which 
no  payments  are  made  until  after  a  specified  time. 
These  types  will  be  taken  up  after  the  ordinary  annuity 
has  been  studied. 

The  amount  of  an  ordinary  annuity  is  indicated  by  s  n. 
This  must  always  be  carefully  distinguished  from  s ", 
which  is  the  symbol  for  compound  amount. 

Problem :  Find  the  amount  of  an  annuity  of  five  annual 
payments  of  $1.00  each,  accumulated  at  4%  annually. 

CO 


ANNUITIES:   AMOUNT  AND  PRESENT  WORTH  61 

First  we  will  establish  the  value  by  accumulating  the 
payments  singly.  Then  w^e  will  learn  a  much  simpler  way 
of  arriving  at  the  same  result. 

First  payment,  accumulated  4  periods,  $1.16985856 

Second     "  «  3        '^  1.12486400 

Third       «  «  2        «  1.08160000 

Fourth     «  «  1        «  1.04000000 

Fifth        «         ,  cash  1.00 

Total  $5.41632256 

The  better  way  is  this:  The  compound  interest  on  $1.00 

for  5  years  at  4%  annually  is  .2166529.     This  amount 

divided  by  the  single  interest  .04  gives  the  same  result 

as  above,  $5.4163225. 

Remembering  that  s  °  oj  (1  +  i)  °  is  the  symbol  for  com- 
pound amount,  it  is  evident  that  compound  interest  is 
s  ° — 1,  or  (1  +  i) "" — 1.    So  we  have  the 

Formula  for  the  amount  of  an  annuity: 

s°— 1       (l+i)°— 1 

S  n    —   — 

i  1 

Rule :  Divide  the  compound  interest  by  the  interest  for 
a  single  period. 

Tables  of  Sn  for  the  usual  rates  and  times  are  found  in  all 
good  collections  of  tables. 

In  accounting  practise,  whenever  an  annuity  occurs,  a 
schedule  should  be  set  up  at  once  covering  the  entire 
life  of  the  annuity.  This  schedule  serves  first  as  a  check 
on  the  accuracy  of  the  computation,  and  second  as  the 
source  of  the  periodic  Journal  entries  necessitated  by  the 
interest  or  other  elements  of  the  annuity. 

Problem :  To  prepare  a  schedule  of  a  series  of  5  annual 
■payments  of  $1,000.00  each,  improved  at  4%  annually. 
We  have  just  seen  that  the  amount  of  such  an  annuity 


Interest 

Year 

4% 

1 

2 

$  40.00 

3' 

81.60 

4 

124.864 

5 

169.8586 

Totals 

416.3226 

62  MATHEMATICS  FOR  THE  ACCOUNTANT 

of  $1.00  is  $5.4163226.     For  $1,000.00  the  amount  is 
evidently  $5,146.3226,  as  is  proved  by  the  following: 

Schedule  of  Accumulation  of  Annuity  of  Five  Annual 
Payments  of  $1,000.00  at  4%  Annually. 

Total 

Annual        Addition        Amount 

Payment       to  Fund        in  Fund 

$1,000.00  $1,000.00      $1,000.00 
1,000.00     1,040.00        2,040.00 
1,000.00     1,081.60        3,121.60 
1,000.00     1,124.864      4,246.464 
1,000.00     1,169.8586    5,416.3226 
5,000.00    5,416.3226 

Such  a  schedule  should  always  be  footed  down  as  a  proof 
of  the  correctness  of  the  additions.  Total  Interest  plus 
total  Payments  must  equal  the  footing  of  Addition  to 
Fund,  and  also  the  last  item  in  the  Amount  in  Fund 
column.  Also,  each  item  in  Total  Addition  to  Fund 
multiplied  by  1  +  i  must  give  the  next  item. 

If  the  schedule  is  lengthy  it  is  a  wise  precaution  to  prove 
every  five  or  at  most  ten  years.  The  first  method  of  proof 
is  by  testing  footings  as  has  just  been  explained.  A  better 
method  is  to  find  the  amount  which  should  be  in  the  fund 
at  the  time.  The  proper  value  of  Sn  multiplied  by  the 
periodic  payment  must  equal  the  amount  in  the  fund  at 
the  time. 

Annuity  problems  often  require  logarithmic  solution. 
For  this  purpose  the  second  formula  given  is  the  most 
easily  understood. 

Problem:  Find  the  amount  of  an  annuity  of  $1.00  a  year 
for  20  years  at  4M%  annually. 


ANNUITIES:  AMOUNT  AND  PRESENT  WORTH  63 

The  compound  amount  must  first  be  found. 

1  +  i  =  1.0425     nl       .0180760636 

times  20  .361521272 

In     =     s°        =         2.2989062 

subtract  1  1.2989062  compound  interest 

divide  by  single  interest  .0425 
gives  $30.5625 

Effective  rates:  So  far  all  our  calculations  have  been 
based  on  annual  or  nominal  rates.  Sometimes,  however, 
the  effective  rate  is  stated,  but  there  are  several  payments 
a  year,  and  there  are  as  many  conversion  intervals  as 
payments.  There  are  two  methods  of  attacking  this 
problem. 

The  first  method  is  to  reckon  the  value^of  the  periodic 
payment  so  that  nominal  rates  can  be  used.  Suppose 
an  annuity  of  $1.00  at  4%  effective,  payments  to  be  made 
quarterly.  If  we  assume  that  the  quarterly  rate  is  1%, 
the  payment  will  be  slightly  less  than  25  cents,  because 
four  payments  of  25  cents  improved  at  1  %  would  amount 
to  more  than  the  specified  $1.00  at  4%.  Tables  of  the 
value  of  such  periodic  payments  are  sometimes  given. 

Another  method  involves  the  use  of  j  p.  Suppose  an 
annuity  of  $1,000.00  for  three  years,  payments  semi- 
annual, at  3%  effective.  If  we  assume  that  the  paj'ment  is 
$500.00,  we  must  accumulate  at  such  a  rate  as  is  equiva- 
lent to  3%  annually.  The  table  of  jp  for  3%  semi- 
annually gives  .0297783;  that  is,  the  semiannual  rate  is 
1.4889%.  It  is  worth  while  to  show  the  proof  schedule 
in  full.  This  schedule  will  be  presented  in  columnar 
form : 

End  of  first      period,  payment         $500.00 
*      second       "       interest  7.4445 

payment  500.00 

Total  $1,007.4445 


Gl  MATHEMATICS  FOR  THE  ACCOUNTANT 

«      third  "       interest  14.9998 

payment  500.00 

Total  $1,522.4443 

«      fourth        "       interest  22.6676 

payment  500.00 

Total  $2,045.1120 

«      fifth           "       interest  30.4495 

payment  500.00 

Total  $2,575.5615 

"      sixth          "       interest  38.3475 

payment  500.00 

Total  $3,113.9090 

Whenever  tables  can  be  used,  the  value  found  in  the 

tables  should  be  multiphed  by  the  quantity  -;-'      The 

effect  of  this  is  to  cancel  i  out  of  the  denominator  of  the 
formula  for  s  n  and  put  j  p  in  its  place.     A  table  of  the 

values  of  —  is  given  in  all  good  collections  of  tables, 
jp 
In  the  problem  for  which  the  schedule  appears  on  the 
preceding  page,  the  table  value  for  a  3-year  annuity  of 

1  at  3%  annually  is  3.0909.    The  value  of  —  when  i   is 

Jp 
3%  and  p  is  2  is  1.0074446.     The  product  of  3.0909  X 
1.0074446  X  1000  is  3113.91,  as  before. 

Whenever  logarithms  are  used,  the  compound  interest 
should  be  divided  by  j  p  instead  of  i.  In  such  cases  both 
compound  amount  and  j  p  have  to  be  found  by  use  of 
logarithms. 

Problem:  Find  the  amount  of  an  annuity  of  $1.00  foi' 
5  years  at  33^%  effective,  the  annuity  being  payable  semi- 
annually. 


ANNUITIES:  AMOUNT  AND  PRESENT  WORTH  66 

First  we  will  find  the  compound  interest: 

1.0325                  nl  .0138900603 

times  5  .0694503015 

In     =     s°  1.1734116 

less  1  .1734116 
Next  we  will  find  jpi 

1.0325                  nl  .0138900603 

divided  by  2  .00694503015 

In  1.01612 

less  1  .01612 

times  2  =  j  p  .03224 

The  quotient  is  $5.3787 

Formulae  for  use  of  j  p  in  finding  the  amount  of  an 
annuity : 

With  tables :  s  n  times  — 


With  logarithms: 


Jp 
(1  +  i)  "-1 


Jp 
In  both  cases  n  is  the  number  of  years. 

The  present  worth  of  an  ordinary  annuity  is  indicated 
by  the  symbol  an.  We  will  demonstrate  the  formula  for 
finding  the  present  worth  just  as  we  demonstrated  the 
amount. 

Problem:  Find  the  present  worth  of  an  annuity  of  $1.00 
for  5  years  at  4%  annually. 

First     payment,  discounted  1  year      .96153846 
Second         «  «  2     «         .92455621 

Thu-d  «  "  3     «        .88899636 

Fourth         «  «  4     «         .85480419 

Fifth  «  «  5     «        .82192711 

Total  $4.45182233 


66  MATHEMATICS  FOR  THE  ACCOUNTANT 

A  quicker  way  is  this:  The  present  worth  of  $1.00  due  in 
5  years  at  4%  annually  is  .82102711;  therefore  the  com- 
pound discount  is  .17807289.  This,  divided  by  the  single 
interest  ,04,  gives  4.45182225  as  above,  allowing  for  in- 
accuracies in  the  right-hand  digits. 

Remembering  that  v  °  is  the  symbol  for  present  worth, 
1  — V  °  is  the  symbol  for  compound  discount.  So  we  have 
the  formula  for  present  worth  of  an  annuity: 

1 — V  ° 

an  =  — : 

1 

For  logarithmic  computation  this  formula  may  be  better 

stated 

1 


1 


(1  -f  i)  -^ 


1 
Rule :  Divide  the  compound  discount  for  the  given  rate 
and  time  by  the  single  interest. 

Tables  of  a  n  for  the  usual  rates  and  times  are  found  in 
all  good  collections  of  tables. 

The  schedule  in  tabular  form  is  necessary  whenever  such 

a  situation   occurs.     For   the   annuity   of  $1,000.00,   5 

years  at  4%  annually,  the  value  of  a  n  is  $4,451.8225. 

Schedule  of  Amortization  of  a  Debt  of  $4,451.8225  Payable 

in  Annual  Instalments  of  $1,000.00,  with  Interest 

on  Outstandings  at  4%  Annually 


Annual 

Net 

Balance 

Interest 

Instal- 

Amortiza-   Outstanding 

Year 

4% 

ment 

tion 

$4,451.8225 

1 

$178.0729 

$1,000.00 

$821.9271 

3,629.8954 

2 

145.1958 

1,000.00 

854.8042 

2,775.0912 

3 

111.0036 

1,000.00 

888.9964 

1,886.0948 

4 

75.4437 

1,000.00 

924.5563 

961.5385 

5 

38.4615 

1,000.00 

961.5385 

Totals 

$548.1775 

$5,000.00  $4,451.8225 

ANNUITIES:  AMOUNT  AND  PRESENT  WORTH  67 

The  usual  checks  should  be  observed.  Total  Interest 
deducted  from  total  Annual  Payments  leaves  total  Net 
Amortization,  which  is  the  same  as  the  amount  of  the 
debt  at  the  beginning.  If  the  schedule  runs  to  twenty 
years  or  more  it  should  be  tested  every  five  or  ten  years, 
by  computing  a  n  as  of  that  date.  Also,  each  item  in 
Net  Amortization  multiplied  by  1  +  i  gives  the  next  item. 
The  logarithmic  solution  is  difficult,  but  needs  to  be 
clearly  understood,  as  the  situation  is  rather  frequent. 
Problem:  Find  the  present  worth  of  a  15-year  annuity 
of  $1.00  at  43^%  annually. 

The  compound  discount  must  first  be  found. 

1  +  i  =  1.0475  nl       .0201540316 

times  15  .302310474 

colog  9.697689526—10 

In  =  v "         =  .4985282 

from  1  =  .5014718  compound  discount 

divided  by  .0475  =  10.5573 

Effective  rates :  The  changes  in  the  formulae  when  j  p 
must  be  used  are  the  same  as  in  the  case  of  the  amount 
of  an  annuity.    The  formulae  are: 

For  use  with  tables,  a  n  times     ^ 

Jp 

1  — V  ° 
For  use  with  logarithms. 


Jp 

In  both  cases  n  is  the  number  of  years;  but  v "  is  reckon- 
ed at  rate  i. 

The  significance  of  the  quantity  an  is  twofold.  In  the 
first  place  it  is  the  present  worth  of  a  debt  of  1  due 
periodically  for  n  periods.  It  is  the  present  payment 
which  is  equivalent  to  such  a  debt  and  will  extinguish 
it  mth  all  interest  that  will  become  due  on  the  successive 
dates  of  payment. 


68  MATHEMATICS  FOR  THE  ACCOUNTANT 

Another  significance  is  this :  It  is  the  present  investment 
which  will  allow  the  owner  to  withdraw  1  periodically; 
in  this  case  the  interest  due  periodically  is  added  to  the 
fund  remaining  in  the  hands  of  the  trustee  or  banker, 
and  serves  to  lengthen  the  time  required  to  consume  the 
investment. 

Three  special  types  of  annuity  are  common : 
An  annuity  due  is  an  annuity  payable  at  the  beginning 
of  the  period.  Insurance  premiums  and  rentals  were 
mentioned  as  common  instances.  For  such  an  annuity 
the  amount  and  present  worth  are  indicated  by  the 
symbols  Sn  and  an.  Suppose  an  annuity  due  of  $1.00 
annually  for  five  years  at  4%  annually. 

We  will  first  find  the  amount  of  this  annuity: 

First      payment,  accumulated  5  periods,  $1.21665290 

Second 

Third 

Fourth         «  « 

Fifth 

Total  $5.63297546 

From  the  tables  we  find  the  amount  of  an  ordinary  annuity 
for  six  periods  to  be  6.63297546.  This  differs  from  the 
amount  of  the  annuity  due  for  five  periods  by  one  cash 
instalment.  The  reason  is  obvious.  The  ordinary  an- 
nuity for  six  periods  ends  with  a  cash  payment  on  which  no 
interest  is  earned.  The  previous  payments  have  earned 
interest  during  one  to  five  periods.  The  annuity  due  is 
exactly  the  same  excepting  for  the  cash  payment. 

Amount  of  an  annuity  due,  first  rule: 

Find  the  amount  of  an  ordinary  annuity  for  one  more 
period,  and  subtract  one  cash  payment. 

Another  aspect  of  the  case:  Each  paj^ment  of  the  an- 
nuity due  has  been  improved  one  period  more  than  each 


4 

« 

1.16985856 

3 

u 

1.124864 

2 

u 

1.0816 

1 

u 

1.04 

ANNUITIES:  AMOUNT  AND  PRESENT  WORTH  69 

payment  of  an  ordinary  annuity  for  the  same  time.  So  if 
we  take  the  amount  of  an  ordinary  annuity  for  the  same 
time  and  multiply  by  1  +  i  we  shall  have  the  amount 
of  an  annuity  due.    5.41632256  X  1.04  =  5.63297546. 

Formulae  for  these  two  rules: 
First,  s  n  =  s  n  +  1  — 1 
Second:  Sn  =  Sn  times  1  -\-  i 

The  present  worth  of  an  annuity  due  of  1  per  period  for 
five  periods  at  4%  annually  is: 

First      payment,  cash  $1.00 

Second         "  discounted  1  period      .96153846 

Third  «  "  2        "  .92455621 

Fourth         "  «  3        "  .88899636 

Fifth  «  «  4       «  .85480419 

$4.62989522 

Now  the  present  worth  of  an  ordinary  annuity  for  four 
years  is  $3.62989522,  which  differs  from  that  of  the 
annuity  due  by  one  cash  payment.  This  is  because  each 
payment  of  the  ordinary  annuity  for  four  years  is  dis- 
counted from  one  to  four  periods.  The  annuity  due  for 
five  periods  has  an  extra  cash  payment,  made  when  the 
annuity  was  entered  on,  and  this  payment  is  not  dis- 
counted. So  one  way  to  find  the  present  worth  of  an 
annuity  due  is  to  find  the  present  worth  of  an  ordinary 
annuity  for  one  less  period,  and  add  one  cash  payment. 
This  is  the  first  rule.  Another  rule  corresponding  to  the 
second  rule  for  summation  of  an  annuity  due  is  this: 
Multiply  the  pre&snt  worth  of  an  ordinary  annuity  for  the 
same  time  by  1  +  i.  4.55182233  X  1.04  =  4.62989522. 
This  is  evidently  because  an  annuity  due  is  payable  one 
period  nearer  the  time  when  it  is  entered  on,  and  so  each 
payment  is  subject  to  discount  for  one  less  period  than 
the  payments  of  an  annuity  made  at  the  end  of  the  period. 


70  MATHEMATICS  FOR  THE  ACCOUNTANT 

Expressed  as  formulae,  these  rules  are, 
First,  a  n  =  a  n- 1  +  1 
Second,  an  =  an  times  1  +  i. 

A  deferred  annuity  is  one  which  is  not  entered  until  after 
a  certain  time.  The  amount  of  such  an  annuity  is  not 
different  from  the  amount  of  an  ordinary  annuity.  No 
payments  are  made  until  after  the  period  of  deferment 
and  consequently  there  are  no  extra  interest  accumu- 
lations. 

The  present  worth,  however,  is  important  is  some  trans- 
actions. There  are  two  methods  of  finding  the  present 
worth  of  such  an  annuity. 

Problem :  Find  the  present  worth  of  a  five-year  annuity 
of  1  per  year,  deferred  three  years,  on  a  4%  annual  basis. 
At  the  date  on  which  the  annuity  is  entered,  its  present 
worth  is  $4.45182233.  This  amount  must  be  discounted 
to  the  present  time.  Multiplying  by  v^  at  5%  we  find 
4.45182233  X  .88899636  =  $3.95765384. 

Rule :  Multiply  the  present  worth  of  the  annuity  by  the 
present  worth  of  1  for  the  period  of  deferment. 

Another  method,  useful  when  table  values  are  obtain- 
able, is  the  following: 

as        at  4%         =         $6.73274487 

a3        at  4%         =  2.77509103 

Subtracting,  $3.95765384 

Rule:  Subtract  the  present  worth  of  an  annuity  for  the 

period  of  deferment  from  the  present  worth  of  an  annuity 

for  the  total  time  from  the  present  to  the  last  payment. 

Formulae  for  these  two  rules  are: 

For  an  n-period  annuity  deferred  m  periods: 

First,  a  n  times  v  ". 

Second,  a  n  +  m  —  am. 


ANNUITIES:  AMOUNT  AND  PRESENT  WORTH  71 

A  peq)etuity  is  a  series  of  periodic  payments  which  are 
expected  to  continue  indefinitely.  The  dividends  on  a 
share  of  preferred  stock  are  one  instance ;  the  income  from 
a  farm  is  another.  The  amount  of  such  an  annuity  has 
no  meaning,  but  the  present  worth  is  important  in  some 
cases,  such  as  an  endowment  of  a  charitable  institution, 
or  the  financing  of  permanent  improvements. 

By  one  of  the  laws  of  simple  interest,  the  principal  can 
be  found  if  we  know  the  periodic  income  and  rate:  in- 
come divided  by  rate  equals  principal.  If  a  perpetuity 
of  1  annually  is  valued  on  a  5%  annual  basis,  its  present 
worth  is  $20.00.  The  symbol  for  a  perpetual  series  or 
infinite  length  of  time  is  m- 

Rule :   Divide  the  rent  by  the  periodic  rate  of  interest : 

As  a  formula,  a^  =  .— 
1 

The  interest  rate  on  a  given  annuity  can  be  found  ap- 
proximately by  Baily's  formula.  The  only  method  of  find- 
ing the  exact  rate  is  to  assume  a  rate  and  find  the  sum  or 
present  worth  as  the  case  may  be,  and  continue  estimating 
and  computing  until  the  correct  rate  is  found.  Baily's 
formula  always  gives  a  rate  slightly  above  the  correct, 
but  it  is  accurate  enough  for  most  occasions.  It  is  neces- 
sary to  know  the  periodic  payment,  the  term,  and  either 
the  amount  or  the  present  worth.  The  first  step  is  to 
read  from  the  tables  the  nearest  value  of  Sn  or  an  for 
the  given  time.  Indicate  these  by  s'  and  a'.  Indicate 
the  interest  rate  for  s'  or  a'  by  j. 

Baily's  formula  provides  for  finding  a  corrective  which 
we  will  call  q.  Then  the  correct  rate  i  =  j  +  q  or 
i  —']  —  q.    The  formulae  for  q  are 


72  MATHEMATICS  FOR  THE  ACCOUNTANT 

j(s^-s) 

s'-n(l+j)n-» 

j^  (a'  —  a) 


a   —  nv  "  "*^  ' 

Required  the  income  rate  on  an  annuity  of  1  yearly  for 
10  years,  when  the  present  worth  is  8.  In  the  table  of 
an  in  the  10-year  row  we  find  7.91271818  in  the  43^% 
column.    V"  at  43^%  is  .61619874. 

.045(8  —  7.91271818) 
^  ~  7.91271818  —  (10  X  .61619874) 
which  gives  q  =  .00224.     Since  the  value  of  a'  chosen 
was  less  than  a,  the  rate  must  have  been  too  high.    Sub- 
tracting .00224  from  .045  gives  i  =  .04276. 

PROBLEMS  ON  CHAPTER  VII 

Find  the  amount  of  these  ordinary  annuities  of  $1.00 
each: 

73.  12  years  at  4  3/^%  annually. 

74.  20  years  at  5%  nominal  quarterly. 

(That  is,  $.25  quarterly.) 

75.  8  years  at  5%  effective  quarterly. 

76.  9  years  at  4  3/^%  nominal  quarterly. 

Find  the  present  worth  of  these  ordinary  annuities  of 
$1.00  each: 

77.  8  years  at  5%  annually. 

78.  11  years  at  6%  nominal  semiannually. 

79.  14  years  at  5%  effective  semiannually. 

80.  6  years  at  53^%  annually. 

81.  24  years  at  43^%  nominal  semiannually. 

Find   the  amount  of  these  annuities  due  of  $1.00  each: 

82.  6  years  at  5%  nominal  semiannually. 


ANNUITIES:  AMOUNT  AND  PRESENT  WORTH  73 

83.  8  years  at  43^%  nominal  quarterly. 

84.  Find  the  present  worth  of  an  annuity  due  of  $1.00 
a  year  for  15  years  at  4)^%  annually. 

85.  Find  the  present  worth  of  a  30-year  annuity  of  $1.00 
on  a  43/^%  annual  basis,  deferred  10  years.  Solve 
two  ways. 

86.  If  $50.00  is  deposited  at  the  end  of  each  six  months 
for  5  years  in  a  bank  paying  4%  nominal  semi- 
annually, what  will  be  the  amount  credited  to  the 
depositor  at  the  end  of  the  term.    Set  up  a  schedule. 

87.  A  man  buys  a  house  for  $1,000.00  cash  and  $1,000.00 
at  the  end  of  each  year  for  three  years.  On  a  6% 
annual  basis  what  is  the  cash  equivalent  of  the  pur- 
chase price? 

88.  The  present  worth  of  an  ordinary  annuity  of  $1.00 
a  year  for  8  years  on  a  5%  annual  basis  is  given  in 
the  tables  as  $6.4632.    Prove  by  setting  up  a  schedule. 

89.  A  man  on  his  40th  birthday  wishes  to  purchase  a 
10-year  annuity  of  $1,000.00  a  year,  the  first  payment 
to  be  mjade  on  his  65th  birthday.  The  insurance 
company  bases  on  3%  annually.  Omitting  prob- 
ability and  loading,  what  is  the  net  single  premium? 
(If  the  first  payment  is  made  on  his  65th  birthday, 
the  annuity  is  entered  on  the  64th  birthday.  Re- 
member also  that  insurance  premiums  are  prepaid). 

90.  A  concern  wishes  to  move  their  manufacturing  plant 
from  its  present  location  on  leased  property.  The 
lease  has  ten  years  to  run,  and  the  rent  is  payable 
annually  in  advance.  The  annual  rental  is  $15,000.00 
a  year  for  the  first  five  years  and  $16,000.00  a  year  for 
the  last  five  yeare.  What  is  the  cash  value  of  the 
lease  on  a  5%  annual  basis. 

91.  The  sum  of  a  15-year  annuity  of  $1.00  a  year  is  $18.00. 
What  is  the  annual  interest  rate? 


74  MATHEMATICS  FOR  THE  ACCOUNTANT 

92.  A  debt  of  $5,000.00  is  to  be  paid  in  20  annual  pay- 
ments of  $400.00  each.  What  is  the  interest  rate? 
(First  reduce  this  to  a  unit  basis.  The  present  worth 
of  this  annuity  is  $12.50  per  dollar  of  annual  pay- 
ment.) 


CHAPTER  VIII 
SINKING  FUNDS 

At  the  beginning  of  Chapter  VII  we  discussed   the 
amount  that  could  be  accumulated  by  periodic  payments 
of  1  improved  at  a  given  rate  of  interest.   In  that  case  we 
knew  the  periodic  payment  and  computed  the  amount.  A 
more  conmion  type  of  problem  is  that  in  which  we  know 
the  amount  which  is  to  be  accumulated  and  the  time  and 
rate,  and  must  find   the  periodic  payment.     Such  an 
amount  to  be  accumulated  within  a  given  time  is  called 
a  sinking  fund,  and  the  periodic  payment  is  called  the 
sinking  fund  payment.    Such  a  sinking  fund  payment  can 
be  found  by  division  from  the  table  of  s  n.    If  annual  pay- 
ments of  1  will  amount  in  five  years  at  4%  annually, 
to  5.41632256,  what  is  the  annual  payment  necessary  to 
amount  to   1   in  five  years  at  4%  annually?     By  the 
principle  of  division  which  was  mentioned  some  time  ago  — 
that  if  a  product  and  one  factor  are  known,  the  other 
factor  can  be  found  by  division — the  sinking  fund  pay- 
ment to  amount  to  1  is  evidently  1  divided  by  5.41632256, 
or  .18462711. 

In  general,  if  periodic  payments  of  1  accumulated  for 
n  periods  amount  to  s  n,  then  to  accumulate  1  the  periodic 
payment  must  be  1  divided  by  Sn,  or,  expresssed  as  a 

fraction,  — 

Sn 

Tables  of  this  quantity  are  given  in  most  collections.  In 

some  collections,  however,  tables  of  another  quantity, — 

an 

75 


76  MATHEMATICS  FOR  THE  ACCOUNTANT 

are  given.    In  such  cases,  as  will  be  explained  in  Chapter 

X,  —  can  be  found  by  subtracting  i  from  the  table  value 

of-i 

aa 

Since  —  is  the  reciprocal  of  Sn,  the   formula    must  be 

the  reciprocal  of  the  formula  for  sn.     The  formula  is 
1  i  i 


-  or 


Sn  S°—  1  (l+i)°—  1 

This  is  for  use  in  logarithmic  computation. 

Rule:  Divide  the  single  interest  by  the  compound 
interest. 

The  following  problem  and  schedule  will  illustrate  the 
principles  of  sinking  fund  mathematics: 

Problem :  What  is  the  annual  sinking  fund  contribution 
necessary  to  provide  a  sum  at  the  end  of  five  years  to 
retire  $10,000.00  bonds,  on  a  4%  annual  basis.  The 
table  value  is  .18462711;  multiphed  by  10,000  gives  the 
annual  payment,  $1,846.2711. 

This  amount  can  be  proved.  The  amount  of  five  annual 
payments  of  1  improved  at  4%  annually  is  given  in  the 
tables  as  5.41832256;  and  multiphed  by  the  amount  of 
the  annual  payment,  $1,846.2711,  the  product  is  about 
10,000. 

When  such  a  sinking  fund  is  begun  it  is  usual  and  ad- 
visable to  build  a  schedule  covering  the  entire  life  of  the 
fund.  The  schedule  for  the  above  bonds  is: 


Year 

1 
2 
3 
4 
5 

Interest 
at  4% 

$  73,8508 
150.6557 
230.5328 
313.6050 

Totals 

768.6443 

SINKING  FUNDS  77 

Schedule  for  Sinking  Fund  to  Redeem  $10,000.00  5% 
Bonds,  Accumulated  at  4%  Annually 

Annual  Total  Amount 

Instal-       Accumula-  in 

ment  tion  Fund 

U,846.2711  $1,846.2711  $1,846.2711 
1,846.2712  1,920.1220  3,766.3931 
1,846.2811  1,996.9268  5,763.3199 
1,846.2712  2,076.8040  7,840.1239 
1,846.2711  2,159.8761  10,000.0000 
9,231.3557  10,000.0000 

Note  that  on  any  date  the  total  of  Annual  Instalments 
plus  Interest  is  the  total  of  Total  Additions  and  also  the 
Amount  in  the  Fund  as  of  that  date.  Also  that  each  Total 
Addition  multiplied  by  1  +  i  gives  the  next  Total  Addi- 
tion. If  the  schedule  runs  for  a  long  time,  the  amount 
in  the  fund  ought  to  be  proved  just  as  in  the  other 
schedules  which  have  been  given.  We  have  shown  that 
the  amount  in  the  fund  on  any  date  can  be  computed 
in  advance  by  multiplying  the  sinking  fund  factor  by 
the  amount  of  an  annuity  for  the  given  number  of  periods. 
In  the  above  case  the  amount  at  the  end  of  the  third  year 
will  be  $1,846.2711  multiplied  by  Sn  for  three  years: 
$1,846.2711  X  3.1216  =  $5,763.3198. 

This  bond  problem  is  unusual  in  that  we  have  assumed 
annual  interest  payments,  whereas  the  usual  bond  pays 
interest  semiannually.  The  general  subject  of  bond  re- 
demption will  be  reserved  for  the  chapter  on  bonds,  at 
which  time  the  method  of  handling  semiannual  interest 
payments  will  be  fully  discussed. 

The  computation  of  the  sinking  fund  payment  for  effec- 
tive rates,  unusual  rates  which  are  computed  by  logar- 
ithms, and  annuities  due  will  now  be  considered.    The 


78  MATHEMATICS  FOR  THE  ACCOUNTANT 

situations  are  in  general  the  exact  reverse  of  those  treated 
in  the  chapter  on  s  n. 
Effective  rates:  The  formula  is 

Sn  Jp 

Rule :  Divide  the  sinking  fund  payment  for  the  time  in 
years  by  the  effective  rate  factor. 

Problem:  What  is  the  monthly  payment  required  to 
accumulate  SI, 000. 00  in  5  years  at  5%  effective,  con- 
vertible monthly? 

.1809748  ^  1.0227148  =  .1769563  the  yearly 

charge  per  dollar. 
.1769563  X  1,000  -  12  =  $14.7463. 
Problem  involving  logarithms: 

The  formula  has  already  been  explained: 

R=  ^— 

Sn — 1 

Problem :  What  is  the  semiannual  payment  into  a  sink- 
ing fund  to  accumulate  $200,000.00  in  25  years  at  4^% 
nominal,  convertible  semiannually. 


IH-  i     =     1.02125 

nl      .0091320695 

times  50 

.456603475 

In     =     s°     = 

2.8615623 

less  1 

1.8615623 

divided  into  .02125 

.01141514 

times  200,000     = 

2283.028 

Annuity  due:    The  formula  is 

R  =  --  ^  (1  + 

i) 

Sn 

Rule:  Divide  the  sinking  fund  factor  for  an  ordinary 
annuity  by  the  compound  interest  ratio. 

Problem:  What  payment  beginning  now  and  continuing 
through  five  years  from  date  will  accumulate  $1,000.00 
at  4>^%  annually. 


SINKING  FUNDS  79 

This  is  an  annuity  due  of  six  payments. 

—  is  given  in  the  tables  as  .14887839 

divided  by  1.045     =  .1424673 

times  1,000     =  $142.4673 

To  determine  the  time  required  to  accumulate  a  stated 
amount:  let  K  be  the  amount  and  R  the  periodic  pay- 
ment.   The  formula  is 

Ki 


log  [l  +  ^] 


R 
n  =  — ; ; r; — 

log  (1  +  i) 

Rule:  Multiply  the  amount  to  be  accumulated  by  the 
periodic  rate  and  divide  by  the  rent;  add  the  result  to  1 
and  find  the  log.  Divide  this  log  by  the  log  of  the  com- 
pound interest  ratio.     (Six-place  log  is  sufficient.) 

Problem:    In  what  time  can  a  fund  of  S10,000.00  be 
accumulated  by  semiannual  payments  of  $500.00  improv- 
ed at  5%  nominal,  semiannually? 
K  =  10,000 


i     = 

.025 

product 

250 

divided  by  500 

.5 

add  1 

1.5 

nl 

.176091 

1  -1-  i  =  1.025 

nl 

.010724 

quotient 

16.42  semiannual  periods. 

PROBLEMS 

ON  CHAPTER  VIII 

93.  What  is  the  semiannual  payment  into  a  sinking  fund 
which  is  to  be  accumulated  at  5%  nominal,  con- 
vertible semiannually,  to  redeem  $50,000.00  of  bonds 
at  maturity,  25  years  from  date? 

From  "  Mathematical  Theory  of  Investment,"  by  Professor  Ernest  Brown  Skinner. 


80  MATHEMATICS  FOR  THE  ACCOUNTANT 

94.  A  debt  of  $250,000.00  is  to  be  paid  in  one  sum  at 
the  end  of  20  years.  The  sinking  fund  is  to  be  accumu- 
lated at  4%  annually.  How  large  is  the  fund  at  the 
end  of  ten  years?    (Schedule  is  not  necessary.) 

95.  What  sum  must  be  deposited  quarterly  in  a  bank 
which  pays  4%  nominal  quarterly  to  amount  to 
$1,000.00  in  10  years? 

96.  In  what  time  will  $50.00  deposited  semiannually  in 
a  bank  which  pays  4%  nominal  semiannually  amount 
to  $1,000.00? 

97.  What  sum  paid  at  the  end  of  each  month  to  an  in- 
surance company  which  accumulates  at  4%  effective 
will  amount  to  $1,000.00  in  10  years? 

98.  What  would  be  the  monthly  payment  in  problem 
97  if  made  in  advance? 

99.  What  will  be  the  annual  payment  into  a  sinking  fund 
accumulated  at  4%  annually  to  extinguish  a  debt  of 
$225,000.00  at  the  end  of  25  years.  Build  a  schedule 
for  the  first  ten  years  and  prove  the  amount  in  the 
fund  at  the  end  of  the  tenth  year. 


CHAPTER  IX 
VALQATION  OF  ASSETS 

One  of  the  commonest  purposes  for  the  building  up  of 
a  sinking  fund  is  the  replacement  of  an  asset.  Such  a 
fund  is  called  a  reserve  fund  for  depreciation.  When 
an  asset  is  purchased,  its  probable  effective  life  and 
replacement  cost  must  be  estimated.  The  phrase  effective 
life  is  used  because  in  any  modern  plant  an  asset  is 
scrapped  long  before  it  has  reached  the  limit  of  its  work- 
ing power.  This  is  due  to  obsolescence,  the  loss  of 
effectiveness  in  the  face  of  more  modern  methods  and 
machines.  This  aspect  of  the  replacement  problem  must 
be  kept  in  mind  when  the  term  of  the  sinking  fund  is 
set,  so  that  the  reserve  may  be  sufficient  in  amount  at  a 
fairly  early  date.  Also  the  replacement  cost  will  be  much 
higher  than  the  original  cost ;  how  much  higher  is  usually 
difficult  to  estimate.  It  is  customary  also  to  set  an 
arbitrary  value  on  the  wornout  machine  as  scrap.  The 
cost,  less  scrap  value,  if  any,  is  called  the  wearing  value ; 
and  it  is  this  wearing  value  which  must  be  accumulated 
during  the  effective  life  of  the  machine. 

The  reserve  fund  then  is  a  sinking  fund  which  must  be 
accumulated  during  the  life  of  the  machine.  The  amount 
of  the  fund  is  replacement  cost  less  scrap  value,  and  the 
term  is  the  effective  life  of  the  machine.  The  replace- 
ment cost  is  indicated  by  C;  the  scrap  value  by  S;  the 
life  by  n ;  and  the  wearing  value  by  W.  W  =  C  —  S,  and 
the  formula  for  the  annual  depreciation  charge  is 

D  =  WX  — 

Sd 

81 


82  MATHEMATICS  FOR  THE  ACCOUNTANT 

Rule:  Multiply  the  wearing  value  by  the  sinking  fund 
factor  for  the  given  time  and  rate. 

Problem:  Determine  the  annual  depreciation  charge 
against  an  asset  which  will  cost  $1,000.00  to  replace  less 
a  scrap  allowance  of  $50.00  at  the  end  of  5  years,  on  a 
33^%  annual  basis. 

W  =  $950.00 
n  =        5 

-i-  =  .18648137 

Sn 

D  =  $177.1573 

A  schedule  should  be  set  up  at  the  time  the  asset  is 
acquired;  one  form  is  as  follows: 


Interest 

Annual 

Total 

Total 

Year 

on  Fund 

Payment 

Addition 

Fund 

1 

$177.1573 

$177.1573 

$177.1573 

2 

$    6.2005 

177.1573 

183.3578 

360.5151 

3 

12.6181 

177.1573 

189.7754 

550.2905 

4 

19.2602 

177.1573 

196.4175 

746.7080 

5 

26.1347 

177.1573 

203.2920 

950.0000 

$  64.2135     $885.7865     $950.0000 

Notice  that  the  total  Annual  Charge  plus  the  total 
Interest  is  the  Wearing  Value.  Also  that  the  Total 
Addition  for  any  year  multiplied  by  1  +  i  gives  the 
Total  Addition  for  the  next  year.  From  the  schedule 
the  periodic  Journal  entry  is  made,  and  the  amount  in 
the  reserve  at  any  time  can  be  read  from  the  Total  Fund 
column.  This  amount  can  be  verified  at  any  time  by 
multiplying  the  annual  charge  by  Sn.  The  total  of  the 
a})ove  fund  at  the  end  of  the  third  year  is  $177.1573  X 
3.106225  =  $550.2903. 


VALUATION  OF  ASSETS  83 

Valuation  of  a  plant  as  a  whole:  In  a  manufacturing 
plant  the  wearing  values  and  terms  of  effective  life  of  the 
machines  vary  widely,  yet  it  is  important  to  arrive  at 
some  method  of  considering  the  plant  as  a  whole.  The 
depreciation  charge  on  each  machine  or  class  of  machines 
must  be  computed  separately,  and  the  results  combined, 
as  in  this  problem: 

Problem :  On  a  4%  annual  basis  find  the  annual  charge 
for  depreciation  against  the  following  electrical  power 
plant : 


Wearing 

Annual 

Part 

Life 

Value 

Charge 

Building 

40 

$  8,000.00 

$  84.1879 

Engine 

20 

3,500.00 

117.5361 

Boiler 

16 

2,000.00 

91,6400 

Dynamo 

18 

7,000.00 

272.9533 

$20,500.00 

$566.3173 

The  depreciation  charge  against  a  plant  as  a  whole  can 
conveniently  be  indicated  by  D'. 

Starting  with  this  annual  charge  on  the  whole  plant, 
four  important  factors  can  be  worked  out  :j 

The  rate  of  depreciation,  d,  is  the  ratio  of  the  periodic 

charge  to    the  wearing  value   at  the  beginning;    for  this 
plant  it  is  566.3173  -^  20,500  -  2.7625%. 

The  composite  life  of  a  plant  is  the  time  within  which 
the  total  of  the  additions  to  the  depreciation  fund  or 
funds  will  accumulate  to  the  original  wearing  values. 
This  is  exactly  the  time  problem  of  page  79: 


n  = 


log  (1  +  i) 


84  MATHEMATICS  FOR  THE  ACCOUNTANT 

Rule :  Divide  i  by  the  rate  of  depreciation,  add  one,  and 
find  the  log;  divide  by  the  log  of  1  +  i- 

This  notion  is  important  when  a  bond  issue  is  secured 
by  a  mortgage  on  a  plant  and  equipment.  Such  bonds 
should  never  be  issued  for  term  longer  than  the  composite 
life  of  the  plant,  and  to  provide  a  proper  margin  of  safety 
the  time  should  be  considerably  shorter.  For  the  above 
plant 

—  =  1.448 

d 

add  1  and  find  log  .388811 

1.04  nl  .017033 

quotient  22.83  years 

The  wearing  value  of  an  asset  at  any  time  is  the  original 
wearing  value  less  the  amount  in  the  reserve  at  that 
time. 

The  condition  percent  of  a  plant  at  any  time  is  the  ratio 
of  the  wearing  value  at  that  time  to  the  original  wearing 
value. 

Other  methods  of  reckoning  depreciation: 
By  allowing  interest  on  the  wearing  value  of  the  asset 
at  the  time.    The  formula  is 

D  =  (Cs"-S)  X  - 

Sn 

Rule:  Multiply  the  cost  by  the  compound  amount 
factor,  and  subtract  the  scrap  value;  multiply  the  result 
by  the  sinking  fund  factor. 

The  composition  of  this  formula  is  easily  understood. 
The  original  cost  multiplied  by  the  compound  amount 
shows  the  capital  tied  up  in  the  machine  during  its  entire 
life.     Of  this  a  small  part  will  be  returned  as  a  scrap 


VALUATION  OP  ASSETS 


85 


allowance.  Deducting  scrap,  we  have  a  loss  of  capital 
which  must  be  spread  over  the  life  of  the  machine  by 
the  sinking  fund  method.  The  objection  to  this  method 
is  that  interest  ought  to  be  reckoned  on  the  original 
cost  if  it  is  to  be  reckoned  at  all.  For  the  asset  con- 
sidered at  the  beginning  of  this  chapter,  the  annual  charge 
by  this  method  is 

1,000  X  1.18768631  less  50  X  .18648137  =  $212.1573 
The  schedule  is 


Net 

Reduction 

ear 

Book 

Interest 

Annual 

in 

Value 

33^% 

Charge 

Book  Value 

1 

11,000.00 

$35,0000 

$212.1573 

$177.1573 

2 

822.8427 

28.7995 

212.1573 

183.3578 

3 

639.4849 

22.3819 

212.1573 

189.7754 

4 

449.7095 

15.7398 

212.1573 

196.4175 

5 

253.2920 

8.8653 

212.1573 

203.2920 

$50.0000     $110.7865  $1,060.7865     $950.0000 

The  phrase  net  book  value  indicates  the  difference  be- 
tween the  cost  and  the  amount  in  the  reserve.  It  is 
difficult  to  make  proper  Journal  entries  from  such  a 
schedule  because  the  asset  should  always  be  carried  on 
the  books  at  cost.  Yet  the  interest  must  be  reckoned  on 
the  net  book  value,  else  this  method  of  charging  for 
depreciation  cannot  be  used. 

Fixed  percentage  on  a  diminishing  book  value:  The 
formula  is 


r  =  1 


^  c 


86  MATHEMATICS  FOR  THE  ACCOUNTANT 

Rule :  Subtract  the  log  of  C  from  the  log  of  S;  divide  the 
result  by  n;  find  the  antilog  and  subtract  from  1.  For  the 
asset  we  have  been  considering 

S  nl  1.698970 

C  nl  3.000000 

subtracting  8.698970—10 

divided  by  5  9.739794—10 

In  .54928 

from  1  .45072 

or  45.072% 

each  year  of  the  net  book  value  at  the  beginning  of  that 
year.    The  schedule  is 

Net  Book        Depreciation       Amount  in 
Year  Value  45.072%  Reserve 

1  $1,000.00  $450.72  $450.72 

2  549.28  247.5714  698.2914 

3  301.7086  135.9861  834.2775 

4  165.7225  74.6944  908.9719 

5  91.0281  41.0281  950.0000 

$50.0000  $950.0000        13,842.2608 

Hatfield,  in  his  "Modern  Accounting,"  discusses  these 
two  methods  in  detail.  He  says  that  the  best  arguments 
for  this  last  method  are,  first  that  the  depreciation  charge 
ought  to  be  heaviest  when  the  machine  is  new  and  less 
in  need  of  repairs,  so  that  year  by  year  as  repair  charges 
increase  they  may  be  compensated  by  lessening  deprecia- 
tion charges;  and  second,  that  investigations  show  that 
a  machine  depreciates  most  rapidly  in  its  first  year  and 
that  the  rate  of  actual  physical  depreciation  decreases 
after  that  time. 

Any  reader  who  is  interested  in  the  more  complicated 
methods  of  reckoning  depreciation,  such  as  the  equal 


VALUATION  OF  ASSETS  87 

annual  payment  method  and  the  unit  cost  method,  should 
read  Saliers'  "Principles  of  Depreciation",  in  uhich  they 
are  treated  fully,  with  schedules  and  diagrams. 

Miscellaneous  formulae  covering  certain  problems  in 
valuation  and  accounting  for  assets: 

Provision  against  the  total  exhaustion  of  wasting  assets 

such  as  mines,  timber  lands,  oil  properties,  and  so  on. 
If  such  assets  were  permanent  or  renewable,  the  provision 
would  take  the  form  of  a  perpetuity,  which  is  valued  by 

the  formula  —  when  R  is  the  annual  income, 
i 

But  since  the  asset  has  a  limited  life  it  is  necessary  to 
provide  also  against  the  time  when  the  rent  will  cease. 
Such  provision  should  take  the  form  of  a  sinking  fund 
payment.  This  payment,  at  a  rate  which  we  may  indicate 
by  i',  must  be  added  to  i  as  an  extra  expense.  The 
formula  for  the  value  of  the  asset  is  then 

V  =  5 

—  (at  rate  i )  +  i 

Sn 

Rule:  Add  the  sinking  fund  charge  at  rate  V  to  the 
income  rate  i;  divide  the  result  into  the  periodic  rent. 

Problem:  Find  the  value  of  a  mine  w^hich  will  net 
$15,000.00  a  year  for  30  years,  if  the  annual  dividend 
rate  is  to  be  6%  payable  annually,  and  the  sinking  fund 
is  to  be  accumulated  at  33^2%  annually. 

—  30  years  at  3^%      =  .01937133 

s 

i  =  .06 

adding  .07937133 

dividing  into  15,000      =  $188,985,110 


88  MATHEMATICS  FOR  THE  ACCOUNTANT 

Viewed  from  another  standpoint  this  solution  gives  the 
amount  of  stock  which  can  safely  be  issued.     This  can 
be  proved.    Basing  on  1,900  shares  of  stock  at  100, 
annual  dividends,  6%  on  $190,000.00.. .  .$11,400.00 
sinking  fund,  $188,985,116 

times  .01937133 3,660.897 

$15,060,897 
which  is  slightly  more  than  the  annual  income  estimated 
at  $15,000.00.  Of  course  it  would  be  conservative  to 
allow  for  contingencies  by  issuing  a  smaller  amount  of 
stock. 

Capitalization  of  assets :  This  is  defined  as  the  first  cost 
plus  cost  of  perpetual  renewals.    The  cost  of  a  perpetuity 

is  - ;  the  rent  of  a  sinking  fund  is  —  .   So  the  product  of 

1  Sn 

these  two  factors  is  the  cost  of  an  indefinite  series  of  re- 
newals occurring  every  n  years.  The  capitalized  cost  is 
the  first  cost  plus  this  cost  of  indefinite  renewals.  This 
problem  occurs  especially  when  endowments  for  hospitals, 
colleges,  and  so  on  have  to  be  valued.  There  are  usually 
items  of  upkeep  which  occur  every  year,  and  these  must 
be  valued  as  perpetuities. 

Problem :  A  philanthropist  desires  to  present  a  building 
to  a  university.  The  building  will  cost  $200,000.00  and 
will  have  a  life  of  50  years.  The  annual  upkeep  will  cost 
not  over  $16,000.00.  What  must  be  the  amount  of  his 
gift? 

First  cost $200,000.00 

Renewals  200,000  X  -    X  .0065502     32751.00 
.04 

16000  400,000.00 

^  ^^^   .04  $632,751.00 


VALUATION  OF  ASSETS  89 

It  is  necessary  to  distinguish  carefully  between 

—   which  is  the  value  of  a  series  of  annual  payments; 
1 

and  -  X  —  which  is  the  value  of  a  series  of  payments  at 
i        Sd 

intervals  of  n  years. 


SUMMARY  OF  FORMULAE  USED  IN 
ACCOUNTING  FOR  ASSETS 

Let  C  be  the  cost,  original  or  replacement; 
S  the  estimated  scrap  value; 
W  the  wearing  value; 
n  the  life  of  a  single  asset; 
n'  the  composite  life  of  an  entire  plant; 
D  the  depreciation  charge  on  a  single  asset; 
D'  the  depreciation  charge  on  an  entire  plant; 
d  the  rate  of  depreciation  of  an  entire  plant,  being 

the  ratio  of  the  total  periodic  charge  to  the 

wearing  value  at  the  begining. 

Annual  depreciation  D  =  W  X   — 

Sn 

Amount  in  fund  at  end  of  r-th  year  Sr  =  D  X  s  n. 
Rate  of  depreciation  d  =   — 

logFi+l] 

Composite  life  of  a  plant  n'  = 

log(H-i) 

Wearing  value  at  end  of  r-th  period  W,  =  W  —  D  X  s  n. 

Wr 

Condition  percent  =  -— ' 
W 


90  MATHEMATICS  FOR  THE  ACCOUNTANT 

Allowing  interest  on  book  value  D  =  (CXs"—  S)X  — 

Vc 

Fixed  rate  on  diminishing  book  value  t  =  1  —y   - 

Value  of  a  wasting  asset  to  pay  dividends  at  rate  i'  when 
R  is  the  annual  output: 

R 


V  = 


—  (at  rate  i')  +  i 

Sn 


Value  of  a  perpetuity,  payable  annually  = 


R 
i 


R        1 

Value  of  a  perpetuity,  payable  every  n  years  =  —  X  — 

1  Sn 

Capitalized  cost  of  an  asset  (this  will  be  treated  in  the 
next  chapter  on  amortization,  but  the  formula  is  given 
here  for  the  sake  of  completeness) : 

X  =  C  X-    X  an+m 
an 

For  most  of  thn=o  formulae  I  am  indebted  to  The  Mathematical  Theory  of  Invest- 
ment, by  Professor  Ernest  Brown  Skinner,  of  the  University  of  Wisconsin. 


PROBLEMS  ON  CHAPTER  IX 

100.  An  asset  having  an  estimated  hfe  of  10  years  and  a 
scrap  value  of  at  least  $500.00  will  cost  $12,000.00 
to  replace.  Set  up  a  schedule  of  the  reserve  by  the 
sinking  fund  method,  basing  on  3i^%  annually. 

101.  An  asset  having  an  estimated  life  of  at  least  8  years 
will  cost  $8,000.00  to  replace  less  an  allowance  of 
$200.00  on  the  old  machine.  Set  up  a  schedule  of 
the  reserve  on  the  fixed  percentage  method. 

102.  An  asset  having  an  estimated  life  of  5  years  and 
estimated  replacement  cost  of  $15,000.00,  less  scrap 


Part 

Life 

A 

15 

B 

25 

C 

45 

D 

30 

E 

8 

VALUATION  OF  ASSETS  91 

allowance  of  $1,200.00  is  to  be  valued  by  the  method 
of  allowing  interest  on  the  book  value  at  the  begin- 
ning of  each  year.  Set  up  a  schedule  on  a  4%  annual 
basis. 

103.  In  an  investigation  of  a  public  utility,  the  various 
parts  were  appraised  as  follows: 

Cost  Scrap 

$150,000.00  $  8,000.00 

65,000.00  3,800.00 

122,000.00  6,750.00 

85,500.00  15,625.00 

8,300.00  650.00 

On  a  3%  annual  basis  compute 

a)  annual    depreciation    charge,    sinking   fund 
method; 

b)  rate  of  depreciation; 

c)  composite  life. 

104.  What  is  the  wearing  value  and  condition  percent  at 
the  end  of  the  7th  year  of  the  first  asset  of  problem 
103? 

105.  Certain  timber  lands  will  yield  a  net  revenue  of 
$8,000.00  a  year  for  35  years.  6H%  stock,  dividends 
annual,  is  to  be  issued,  and  provision  is  to  be  made  for 
annual  payments  to  a  sinking  fund  at  4%  annually. 
Supposing  that  the  land  has  no  residual  value,  what 
is  a  reasonable  purchase  price? 

106.  It  is  estimated  that  a  mine  will  yield  $12,000.00  per 
year  and  will  be  exhausted  in  20  years.  The  stock 
is  to  pay  7%  annually  and  the  sinking  fund  is  to  be 
accumulated  at  43/2%  annually.  Find  the  amount  of 
stock  that  can  be  safely  issued. 

107.  How  much  can  a  railroad  afford  to  spend  in  elimi- 
nating a  grade  crossing  which  is  guarded  by  two 


92  MATHEMATICS  FOR  THE  ACCOUNTANT 

watchmen  at  $650.00  a  year  each.     Assume  4% 
annually. 

108.  What  sum  must  be  set  aside  annually  to  provide  for 
rebuilding  every  25  years  of  a  bridge  costing 
$30,000.00?  Assume  that  money  will  be  worth  4% 
annually. 

109.  On  a  certain  railroad  grade  a  helper  engine  and 
two  crews  are  necessary  at  an  annual  cost  of 
$12,000.00;  a  new  engine  must  be  provided  every 
20  years  at  a  cost  of  $14,000.00.  How  much  can 
the  railroad  afford  to  spend  to  level  the  grade  if 
money  will  be  worth  4J^%  steadily? 

110.  A  man  desires  to  endow  an  art  gallery.  The  building 
will  cost  $75,000.00.  The  life  is  100  years.  The 
annual  cost  for  purchase  and  rental  of  paintings  and 
so  forth  will  be  $50,000.00.  Repairs  and  renewals 
will  cost  $2,000.00  yearly.  The  lighting  system  will 
need  renovating  every  5  years  at  a  cost  of  $1,200.00. 
Upkeep  and  service  will  cost  $4,500.00  a  year. 
Value  the  endowment  on  a  33^%  annual  basis. 


CHAPTER  X 

AMORTIZATION 

In  Chapter  VII  we  discussed  amortization  somewhat. 
It  can  be  defined  as  the  series  of  equal  payments  which 
will  extinguish  a  given  debt,  the  periodic  payments  being 
used  first  to  meet  interest  accrued  on  balances  outstand- 
ing. In  that  chapter  we  learned  that  five  equal  payments 
of  1  will  extinguish  a  debt  of  4.45182233  with  interest  at 
4%.  The  more  common  situation  is  that  in  which  the 
amount  of  the  debt  is  known  and  the  amount  of  the 
periodic  payment  must  be  found.  What,  for  instance,  is 
the  periodic  payment  which  will  extinguish  a  debt  of  1 
in  5  years,  interest  on  outstandings  at  4%  annually? 
Evidently,  if  payments  of  1  will  extinguish  a  debt  of 
4.45182233,  then  the  payment  required  to  extinguish  a 
debt  of  1  is  1  divided  by  4.45182233,  or  .22462711.  And 
the  payment  required  to  extinguish  a  debt  of  $10,000.00 
is  $2,246.2711.    A  schedule  will  prove  this: 

Schedule  of  Amortization  of  a  Debt  of  $10,000.00  in  5 

Equal  Annual   Instalments,   with   Interest   on 

Outstandings  at  4%  Annually 


Balance 

Net 

Out- 

Interest        Annual       Amortiza- 

Year 

standing 

at  4%         Payment           tion 

1 

$10,000.0000 

$400.0000  $2,246.2711  $1,846.2711 

2 

8,153.7289 

326.1492    2,246.2711     1,920.1219 

3 

6,233.6070 

249.3443    2,246.2711     1,996.9268 

4 

4,236.6802 

169.4672    2,246.1712    2,076.8040 

5 

2,159.8762 

86.3950    2,246.2712    2,159.8762 

1,231.3557  11,231.3557  10,000.0000 

93 

94  MATHEMATICS  FOR  THE  ACCOUNTANT 

The  usual  checks  should  be  observed.  Total  Annual 
Payments  less  total  Interest  is  total  Net  Amortization, 
which  is  the  same  as  the  Balance  Outstanding  at  the 
beginning.  Each  Net  Amortization  times  1  +  i  gives  the 
next  amount.  The  balance  outstanding  on  any  interest 
date  is  the  present  worth  of  the  annual  payments  still 
due.  In  the  above  schedule,  the  amount  due  just  after 
the  third  payment  is  the  present  worth  of  the  two  remain- 
ing payments;  $2,246.2711  X  1.88609467  =  S4,236.678. 
If  the  schedule  is  lengthy  it  should  be  proved  every  few 
years  with  this  last  check. 

"To  pass  to  algebra,  since  a  debt  of  a  n  can  be  amortized 
by  a  series  of  n  payments  of  1,  a  debt  of  1  can  be  amortized 
by  a  series  of  n  payments  of  1  divided  by  a  n,  or,  expressed 

1 

as  a  fraction,  — 

an. 
For  logarithmic  computation,  the  formula  is  the  recip- 
rocal of  the  formula  for  a  n, 

J^    ^   _J i 

an  1  — V°         - 1 


(1  +  i)  " 
and  in  the  usual  circumstances,  when  the  given  rate  is 

efifective,  if  we  are  using  table  values,  we  divide :  ■ — ■    -^    — 

1  Jp 

where  n  is  the  number  of  years ;  and  if  we  are  using  logar- 
ithms, we  use  j  p  instead  of  i  in  the  numerator  of  the 
formula,  while  n  in  the  denominator  is  the  number  of 
years.    These  will  be  illustrated  later. 

Now,  if  we  compare  the  interest  columns  of  the  sinking 
fund  and  amortization  schedules,  we  shall  find  an  in- 
teresting relation  between  the  sinking  fund  and  amortiza- 
tion factors.  In  the  sinking  fund  schedule  the  interest  is 
a  credit ;  in  the  amortization  schedule  it  is  a  debit. 


AMORTIZATION 

Amortization 

Sinking 

Year 

Debit 

Fund  Credit 

1 

$400.00 

2 

326.1492 

$  73.8508 

3 

249.3443 

150.6557 

4 

169.4672 

230.5328 

5 

86.3950 

313.6050 

95 


Id  each  year  the  sum  of  the  two  items  is  4%  of  $10,000.00. 
Or,  a  debit  of  $400.00  set  against  each  credit  to  the  sinking 
fund  results  in  the  corresponding  debit  to  amortization. 

This  shows  why  the  tables  do  not  include  any  tab'e  of  — 

an' 

for  that  quantity  is  always  greater  than  -^  by  the  annual 

Sn 

interest.  This  fact,  stated  on  the  basis  of  a  unit,  as  usual, 
gives  the  formula:  .  . 

—  =  -  +  i 

an  Sn 

Rule :  To  find  the  amortization  factor  for  any  time  and 
rate,  read  the  table  value  for  the  corresponding  sinking 
fund  factor  and  add  the  interest  rate. 

In  the  sinking  fund  table  for  4%  when  n  =  5  we  find 
.18462711;  adding  .04  we  have  the  amortization  factor 
.22462711,  the  same  amount  we  found  on  the  previous 
page  by  division. 

Logarithmic  solution :  Find  the  rent  of  an  amortization 
of  a  debt  of  1  in  10  semiannual  payments  at  5}/2%  nominal, 
convertible  semiannually : 

1  +  i  =  1.0275  nl     .0117818305 

times  10  .117818305 

colog  =  log  of  v"     9.882181695—10 

antilog  =  v°  .7623977 

from  1  .2376023  the  compound  discount 

divide  into  .0275  =  .11574 


96  MATHEMATICS  FOR  THE  ACCOUNTANT 

Rule:    Divide  the  interest  rate  by  the  compound  dis- 
count. 

Effective  rates:  What  is  the  monthly  payment  required 

to  amortize  a  debt  of  $500.00,  beginning  at  the  end  of  one 

month  and  continuing  for  5  years,  interest  at  5%  effective. 

1 


an 
i 


.2309748 
at  5%,  p  =  12,  =        1.0227148 


Jp 

quotient       =  .2258448 

times  500     =  112.9224 

divided  by  12     =  $9.41 

Annuity  due:   Since  the  formula  for  the  present  worth 
of  an  annuity  due  is  En  X  (1  +  i),  it  is  evident  that  the 

amortization  factor  payable  in  advance  is  —  ^  (1  +  i). 

an 

Problem:    A  debt  of  1  is  to  be  amortized  in  5  annual 

payments,  the  first  payment  to  be  made  immediately. 

What  is  the  annual  payment  on  a  6%  annual  basis? 

—  =  .2373964 

dividing  by  1.06     =  .223959 

Four  special  problems  in  amortization  are  important: 
The  amount  due  after  any  number  of  payments  have 
been  made:  This  has  been  mentioned  already,  and  was 
explained  as  the  present  worth  of  the  payments  still  due. 
This  is  true  inmaediately  after  a  payment  has  been  made. 
If,  however,  the  amount  due  immediately  before  a  date 
of  payment  is  in  question,  the  payments  still  due  con- 
stitute an  annuity  due  and  must  be  valued  according  to 
the  formula  for  such  an  annuity. 

Problem:   A  debt  of  $10,000.00  is  to  be  amortized  over 
a  period  of  10  years  by  annual  payments,  interest  at  5% 


AMORTIZATION  97 

annually.  On  the  date  of  the  seventh  payment  it  is 
desired  to  liquidate  the  balance  in  one  amount.  What  is 
the  balance? 


.12950458 


_1 

aio 

times  10,000  1,295.0458  annual  payment 

as  2.72324803 

product  3,526.7309 

add  7th  payment  1,295.0458 

Amount  due  $4,921.7767 

Or,  we  might  have  multiplied  3.549505  X  1.05  X 
1,295.0458,  since  3.5459505  is  the  present  worth  of  an 
ordinary  annuity  for  4  periods  at  5%. 

The  time  required  to  amortize  a  given  debt:  Let  A  be 
the  debt  and  R  the  rent,  or  periodic  payment.  The 
formula  is* 

colog[l-|] 

""  -      log  (1  +  i) 
Rule :  Multiply  the  amount  by  the  interest  rate,  divide 
by  the  rent,  subtract  from  1,  find  the  colog;  divide  by  the 
log  of  the  interest  ratio. 

Problem:  In  what  time  can  a  debt  of  $1,000.00  be 
liquidated  by  annual  payments  of  $80.00,  interest  at  5% 
annually? 


A 

= 

$1,000.00 

R 

= 

80.00 

i 

= 

.05 

Ai 

= 

50 

divided  by  80 

.62J 

from  1 

.37i 

From  "  Mathematical  Theory  of  Investment,"  by  Professor  Ernest  Brown  Skinner. 


98  MATHEMATICS  FOR  THE  ACCOUNTANT 

nl  9.574031—10 

colog  .425969 

1.05  nl  .021189 

quotient  20.1 

When  the  rate  is  effective,  j  p  is  used  in  place  of  i  in  the 
numerator  (multipHed  by  A),  but  R  is  the  annual  pay- 
ment, and  i  is  the  annual  rate. 

Problem :  In  what  time  will  a  debt  of  $300.00  be  amor- 
tized by  quarterly  payments  of  $10.00,  beginning  3 
months  from  date,  interest  at  33^2%  effective? 

A  =  $300.00 

R  =  40.00 

j  p  when  p  =  4  .0345498 

A  times  j  p  10.36494 

divided  by  40  .2591235 

from  1  .7408765 

nl  9.869746—10 

colog  .130254 

1.035  nl  .01494 

quotient  8.72 

or  8^  years  and  a  small 

balance  due. 
The  increased  cost  of  lengthening  the  life  of  an  asset 
Let  C  be  the  cost  of  the  present  asset,  andC  the  cost  of 
the  longer-lived  asset.  Let  n  be  the  present  life,  and  m 
the  added  life,  so  that  n  +  m  is  the  total  life  of  the  new 
asset.  Now,  if  we  multiply  the  present  cost  by  the 
amortization  factor,  we  shall  have  the  annual  cost  of  the 
present  asset.  And  if  we  multiply  that  by  the  present 
worth  of  an  annuity  of  1  for  the  lengthened  term  of  the 
new  asset  we  shall  have  the  amount  which  we  can  afford 
to  pay  for  the  new  asset.     Expressed  as  a  formula* 

From  "Mathematical  Theory  of  Investment,"  by  Professor  Ernest  Brown  Skinner. 


AMORTIZATION  99 

1 

C  =  C  X X  an  +  m 

an 

Problem :  If  a  piece  of  equipment  costs'^!  100.00  and  has 
a  life  of  3  years,  how  much  can  be  paid  for  a  better  piece 
of  equipment  which  will  last  5  years,  on  a  4%  annual 
basis? 

C  =  $100.00 

—  =  .36034854 

aa 

product  36.034854  annual  cost 

Ef  =  4.45182233 

product  S160.42 

That  is,  the  equipment  costs  us  .36034854  a  year,  for 

any  term  of  years.    The  present  worth  of  a  5-year  article 

is  that  amount  times  the  present  worth  of  an  annuity  of 

1  for  5  years. 

The  amordzation  of  a  series  of  bonds  will  be  discussed 

in  the  next  chapter. 

PROBLEMS  ON  CHAPTER  X 

111.  A  man  buys  an  estate  valued  at  S8,000.00,  agreeing 
to  pay  cash  S2,000.00  and  the  balance  in  5  equal 
annual  payments  with  interest  on  outstandings  at 
5%  annually.    Set  up  a  schedule  of  the  payments. 

112.  A  man  borrows  $3,000.00  at  5%  annually,  agreeing 
to  pay  $300.00  annually.  By  the  use  of  a  formula 
determine  the  life  of  the  debt.  What  is  the  amount 
due  after  the  last  full  period.  Solve  without 
schedule. 

113.  What  is  the  annual  rent  to  amortize  a  debt  of 
$1,000.00  in  10  years  at  4^%  annually. 

1 14.  What  quarterly  payment  continued  three  years  will 
pay  a  debt  of  $200.00  with  interest  at  4%  effective. 


100  MATHEMATICS  FOR  THE  ACCOUNTANT 

115.  What  is  the  annuity  required  to  amortize  a  debt  of 
SI, 500.00  with  interest  at  3^%  annually,  in  5  pay- 
ments, the  first  payment  to  be  made  at  once. 

116.  Set  up  a  schedule  for  problem  115. 

117.  A  debt  of  $3,000.00  with  interest  at  43^%  annually 
is  to  be  amortized  in  10  annual  payments,  beginning 
one  year  from  date.  If  the  payments  are  made 
promptly  for  five  years,  what  amount  will  liquidate 
the  balance  due  on  the  date  of  the  sixth  payment. 

118.  A  city  incurs  a  debt  of  $150,000.00.  From  a  mathe- 
matical standpoint  which  is  better:  to  amortise  in 
20  equal  annual  payments  at  Q}-i%  annually,  or  to 
accumulate  a  sinking  fund  by  annual  payments  im- 
proved Sit  4:%  annually,  paying  interest  on  the  face 
of  the  debt  meanwhile  at  5%  annually. 

119.  A  man  is  offered  $12,000.00  a  year  payable  in  ad- 
vance on  a  99-year  lease  or  cash  $225,000.00  for  a 
piece  of  land.  Which  is  better  for  him  on  a  5% 
annual  basis? 

120.  On  a  43^%  annual  basis  what  amount  can  be  spent 
in  improving  a  piece  of  apparatus  which  now  costs 
$25.00  and  has  a  life  of  3  years  so  that  it  may  last 
5  years? 

121.  On  a  5%  annual  basis  what  can  a  company  afford  to 
spend  in  improving  a  machine  which  now  costs  $85.00 
and  has  a  life  of  6  years  so  that  it  may  last  9  years? 


CHAPTER  XI 
VALUATION  OF  BONDS 

A  bond  is  a  promise  to  pay  a  certain  sum,  usually 
$1,000.00  or  $100.00,  after  a  certain  number  of  years 
from  the  date  of  issue.  The  commonest  exception  to  this 
statement  is  the  issue  of  a  series  redeemable  by  drawings. 
It  also  promises,  during  the  life  of  the  bond,  to  pay 
interest  at  a  fixed  rate,  called  the  coupon  rate;  such 
interest  is  usually  paid  semiannually.  In  this  book  this 
coupon  rate  will  be  indicated  by  c;  actuarial  writers 
usually  use  j,  because,  as  we  shall  see,  this  coupon  rate 
is  really  a  nominal  rate  and  does  not  indicate  the  income 
rate  realized  by  the  holder.  Bonds  are  commonly  pur- 
chased either  above  par,  ''at  a  premium";  or  below  par, 
"at  a  discount".  The  amount  of  premium  or  discount 
may  depend  on  a  number  of  things;  but  if  the  bond  is 
"gilt-edged"  and  purchased  for  investment  lather  than 
for  speculation  the  price  depends  chiefly  on  the  income 
rate  expected,  and  after  that  on  the  value  at  which  the 
bond  will  be  redeemed  and  the  time  to  elapse  before 
redemption.  The  income  rate,  which  we  will  call  i,  is 
independent  of  c;  in  fact  it  is  usually  not  calculated  until 
after  purchase  and  when  the  bond  is  brought  on  the  books. 
If  the  coupon  and  income  rates  are  the  same  the  bond  is 
bought  at  par.  If  the  purchaser  is  satisfied  that  his 
income  shall  be  less  than  the  nominal  coupon  rate, 
the  bond  will  be  purchased  at  a  premium.  If  the  pur- 
chaser wishes  his  income  rate  to  be  above  the  coupon 
rate,  he  will  purchase  at  a  discount. 

The  value  of  a  bond  depends  then  on  the  redemption 
value  and  the  relation  between  the  coupon  and  income 

101 


102  MATHEMATICS  FOR  THE  ACCOUNTANT 

rates.  Since  coupons  are  usually  payable  semiannually 
it  is  customary  to  value  bonds  on  a  semiannual  basis, 
unless  the  bond  is  stated  to  pay  annual  or  quarterly 
coupons.  Let  us  value  a  bond  by  valuing  a)  the  present 
value  of  the  redemption  price,  and  b)  the  present  value 
of  the  coupons. 

Problem:  What  is  the  price  of  a  7%  25-year  bond, 
redeemable  at  par  $1,000.00,  to  yield  6%. 

Since  the  valuation  is  to  be  on  a  semiannual  basis: 
n  =  50 

c  =  .035 

i  =  .03 

Present  worth  of  the  redemption  value  is 
1,000  X  v^o  at  3%,  which  is 
1,000  X  .22821708  =  $228.1071 

Present  worth  of  50  coupons  of 
$35  each  is 
35  X  25.729764  =  900.5417 

Total $1,128.6488 

This  establishes  the  correct  value  of  the  bond.  The  best 
method  of  valuation  is  the  formula  of  Mr.  Makeham,  as 
follows : 

Let  C  be  the  redemption  value  (cash) ; 
K  the  present  worth  of  C ; 
c  the  coupon  rate  (Makeham  uses  j) ; 
i  the  income  rate; 
A  the  present  value  of  the  bond. 

The  formula  is  A  =  K  -^  X  (C  -  K) 

1 

Rule:  Find  the  present  worth  of  the  redemption  value. 
Subtract  this  from  the  redemption  value  and  multiply 
the  result  by  the  coupon  rate  and  divide  by  the  income 
rate.    Add  these  results. 


VALUATION  OF  BONDS  103 

For  the  bond  we  have  just  been  considering: 
K  =  1,000  X  .22810708       =        S228.1071 
C  —  K  =  771.8929 
.035 


.03 


X  771.8929        =    900.5417 


$1,128.6488 

The  reason  for  this  formula  is  this.  If  the  bond  were 
purchased  so  that  coupon  and  yield  rates  were  the  same, 
the  price  would  be  the  same  as  the  redemption  value. 
Since  the  present  worth  of  the  redemption  value  is 
$228.1071,  the  value  of  the  coupons  would  be  the  re- 
mainder of  the  $1,000.00,  or  $771.8929.  But  the  yield 
rate  is  lower  than  the  coupon  rate,  which  shows  that  the 
bond  cost  more  than  par.  It  cost  as  much  above  par  as 
the  coupon  rate  is  above  the  yield  rate,  which  in  this 

.    -035.    , 
case  IS  — r—     In  every  case  the  coupon  rate  is  the  numer- 
ator of  this  fraction. 

It  is  usual  to  set  up  a  schedule  at  the  time  the  bond  is 
brought  on  the  books ;  the  first  ten  periods  of  this  schedule 
are: 

Schedule  of  Amortization  of  7%  25- Year  Bond  of 

the Co., 

Redeemable  at  Par  $1,000.00  on , 

Income  Rate  6% 

Net 
Coupons        Income       Amortiza-         Book 
Period        33^%  3%  tion  Value 

$1,128.6488 

1  $35.00    $33.8595   $1.1405    1,127.5083 

2  35.00     33.8252    1.1748    1,126.3335 

3  35.00     33.7900    1.2100    1,125.1235 

4  35.00     33.7537    1.2463    1,123.8772 


104  MATHEMATICS  FOR  THE  ACCOUNTANT 


5 

35.00 

33.7163 

1.2837 

1,122.5935 

() 

35.00 

33.6778 

1.3222 

1,121.2713 

7 

35.00 

33.6381 

1.3619 

1,119.9094 

8 

35.00 

33.5973 

1.4027 

1,118.5067 

9 

35.00 

33.5552 

1.4448 

1,117.0619 

0 

35.00 

33.5119 

1.4881 

1,115.5738 

350.00  336.9250  13.0750 
There  are  several  checks  to  such  a  schedule,  the  first 
of  which  is  by  the  usual  additions.  If  the  bond  was  bought 
at  a  premium,  total  Coupons  less  total  Income  equals 
total  Net  Amortization;  each  Net  Amortization  multi- 
plied by  1  +  i  gives  the  next.  If  the  bond  was  bought 
at  a  discount,  total  Income  less  total  Coupons  gives  total 
Net  Accumulation;  and  each  Net  Accumulation  times 
1  -1-  i  gives  the  next. 

The  next  check  is  by  valuing  the  bond  again: 

K  =  1,000  X  v^o  = 

1,000  X  .3065568         =        $306.5568 

C-K  =  693.4432 

.035 

— -  X  693.4432  =  809.0171 

$1,115.5739 
Still  another  check  is  by  difTerencing.  This  has  the 
merit  that  it  is  not  in  any  way  similar  to  the  methods 
used  in  valuation,  and  so  is  less  likely  to  repeat  any 
mistake  that  may  have  been  made.  Set  down  in  a 
column  the  successive  Net  Amortizations  (in  the  table 
they  have  been  set  down  in  reverse  order  to  facilitate 
subtraction).  Subtract  each  one  from  the  one  directly 
over  it;  this  is  called  the  first  order  of  differences.  Repeat 
the  operation,  securing  a  second  order  of  differences. 
If  desirable  subtract  once  more.  In  the  accompanying 
schedule  this  is  unnecessary.  It  is  unnecessary  to  pro- 
ceed beyond  the  point  where  the  differences  are  nearly 


VALUATION  OF  BONDS  105 

equal,  some  being  slightly  more  and  some  slightly  less. 
If  any  wide  variation  should  appear  it  would  indicate 
an  error  in  that  region  of  the  schedule. 

1.4881  .0433  .0012 

1.4448  .0421  .0013 

1.4027  .0408  .0011 

1.3619  .0397  .0012 

1.3222  .0385  .0011 

1.2837  .0374  .0011 

1.2463  .0363  .0011 

1.2100  .0352  ,      .0009 

1.1748  .0343 
1.1405 

In  the  second  column  the  differences  are  progressive; 
that  is,  they  decrease  successively.  In  the  third  column 
the  differences  waver.  In  a  short-term  schedule  the 
differences  are  larger  than  in  a  long-term  schedule.  If 
this  were  a  five-year  bond  it  would  probably  be  neces- 
sary to  obtain  a  fourth  column  to  reach  the  point  where 
the  differences  were  nearly  equal. 

When  the  income  rate  is  not  in  the  tables  the  use  of 
logarithms  is  necessary  in  finding  v  °.  The  process  has 
been  demonstrated,  but  will  be  shown  here  again. 

Problem:  Value  a  5%  20-year  bond  redeemable  at 
par  $1,000.00,  to  yield  3.6%. 


c                     = 

.025 

i                   = 

.018 

n                   = 

40 

1  -1-  i  =  1.018    nl 

.0047747778 

times  40 

.30991112 

colog 

9.69008888-10 

In  =  v  °  = 

.    .4898794 

106  MATHEMATICS  FOR  THE  ACCOUNTANT 

K  =  1,000  X  .4898794 

=      $489.8794 
C— K  =  510.1206 

—  X  510.1206       =  708.5008 

^•^  $1,198.3802 

Redemption  at  a  premium  or  discount:  Bonds  are  often 
issued  subject  to  an  agreement  that  they  may  be  redeemed 
above  or  below  par.  This  is  especially  common  when  the 
issuing  company  reserves  the  right  to  redeem  before 
maturity.  Suppose  a  5%  bond  can  be  redeemed  at  110, 
when  par  is  100.  The  income  rate  is  based  on  the  pur- 
chase price,  so  the  redemption  value  has  no  direct  effect 
on  the  income  rate.  But  5%  on  100  is  not  5%  on  110. 
The  coupon  rate  must  be  converted  to  this  new  value 

at  maturity.     X  .05  =  .045454  annually,  or  .022727 

semiannually.     Suppose  this  bond  is  redeemable  in  15 
years,  that  is  n  =  30.    Income  rate  4%: 

K  =  1,100  X  v30at2% 

=  1,100  X  .55207089       =        $607.2780 

C— K  =  492.7220 

.022727 


.02 


X  492.7220  =  559.9113 


$1,167.1893 

Valuation  with  allowance  for  income  tax:  The  writers 
on  actuarial  science  are  forced  to  treat  this  subject  at 
length  and  to  give  complicated  formulae  because  the 
English  income  tax  law  bases  taxation  on  the  net  in- 
come after  accumulation  of  discount  or  amortization  of 
premium.  But  in  this  country  where  the  tax  is  levied  on 
the  coupons  themselves,  the  problem  is  simple.  Consider 
a  20-year  5%  bond  redeemable  at  par  $1,000.00.  The 
purchaser   wishes  to  net   4%  after  taxation,  and   esti- 


VALUATION  OF  BONDS  107 

mates  that  the  tax  will  consume  10%  of  all  his  income. 
Of  every  2.5%  of  income  one- tenth  will  satisfy  the  tax. 
That  means  that  he  will  net  2.25%.  The  problem  then  is 
to  value  a  40-period  23^%  bond  on  a  2%  yield: 

K  =  1,000  X  .4528904         =        $452.8904 

C— K  =  547.1096 

.0225 


.02 


X  547.1096  =  615.4983 


$1,068.3887 

Without  the  allowance  for  tax  the  cost  of  the  bond 
would  have  been  $1,136.7774,  the  difference  being 
68.3887. 

The  proof  of  this  is  very  interesting.  The  holder  of 
this  bond  expects  to  pay  $2.50  of  every  coupon  in  tax. 
The  present  worth  of  a  40-period  annuity  on  a  2%  basis 
is  27.35547924.  The  product  of  27.35547924  X  2.50  is 
$68.3887,  which  was  exactly  the  decrease  in  purchase 
price  due  to  the  allowance  for  taxation. 

Valuation  between  interest  dates:  So  far  we  have  been 
valuing  bonds  on  interest  dates  or  at  least  ex-interest. 
In  practise,  bonds  are  usually  bought  between  interest 
dates,  and  if  the  price  is  accurately  fixed  there  must  be 
not  only  an  allowance  for  interest  due  the  former  holder 
but  also  adjustment  for  discount  or  premium  for  the  part 
of  a  period  already  past.  It  is  usual  to  reckon  the  interest 
and  premium  or  discount  by  simple  interest,  as  follows: 

Problem:  A  7%  25-year  bond,  redeemable  at  par 
$1,000.00,  is  bought  on  a  6%  basis.  The  coupons  are  due 
January  and  July  1.  The  bond  is  bought  March  1. 
What  is  the  price? 

We  found  that  the  price  January  1  v/as  $1,128.6488. 
The  seller  is  entitled  to  the  third  of  the  coupon  which 
has  accrued  and  must  expect  to  lose  the  third  of  the 
premium  to  be  amortized: 


108  MATHEMATICS  FOR  THE  ACCOUNTANT 

Value  January  1  $1,128.6488 

1/3  of  $35.00  $11.6667 

less  1/3  of  3% 

of  book  value  11.2865 .3802 

Subtracting,  value  March  1         $1,128.2686 
Also,  it  often  happens  that  the  coupon  dates  do  not 
coincide  with  the  dates  for  closing  the  books.     In  this 
case  it  is  desirable  to  write  a  schedule  which  will  show 
the  value  of  the  bond  on  the  dates  of  closing. 

Problem:  A  bond  redeemable  for  $1,000.00  on  October 
15,  1916,  bears  interest  at  5%  payable  April  and  October 
15.  The  bond  is  purchased  April  15,  1913,  to  yield  6%. 
Value  the  bond  and  write  a  schedule  which  can  be  used 
as  a  source  of  entries  on  a  set  of  books  which  are  closed 
June  30  and  December  31 : 


Value  April  15 

$968.8486 

5/12  of  3%  of  book 

value 

$12.1106 

5/12  of  coupon 

10.4167          1.6939 

Adding,  value  June  30 

$970.5425 

The  schedule  is : 

Net 

Income 

Accumu- 

Book 

Date               3% 

Coupons 

lation 

Value 

4/15/'13 

$968.8486 

6/30/'13      $12.1106 

$10.4167 

$1.6939 

970.5425 

12/31/'13        29.1163 

25.00 

4.1163 

974.6588 

6/30/'14        29.2398 

25.00 

4.2398 

978.8986 

12/31/'14        29.3670 

25.00 

4.3670 

983.2656 

6/30/'15        29.4980 

25.00 

4.4980 

987.7636 

12/31/'15        29.6329 

25.00 

4.6329 

992.3965 

6/30/'16        29.7719 

25.00 

4.7719 

997.1684 

10/15/'16        17.4149 

14.5833 

2.8316 

1,000.0000 

$206.1514 

$175.0000 

$31.1514 

VALUATION  OF  BONDS  109 

The  usual  checks  should  be  observed.  A  slight  discrep- 
ancy will  usually  appear  in  the  net  accumulation  or 
amortization  for  the  last  partial  period.  In  the  above 
schedule  this  has  all  been  taken  up  by  adjusting  the 
income.  There  are  various  methods  of  eliminating  this 
residue,  as  it  is  called.  It  can  all  be  taken  care  of  during 
the  first  period  or  during  the  last  period.  It  can  be 
spread  equally  over  all  tne  periods.  Or  it  can  be  spread 
over  all  the  periods  proportionately  to  the  net  accumu- 
lations or  amortizations.  By  this  last  method  an  adjust- 
ment per  dollar  is  found,  basing  on  the  total  of  the  net 
column.  This  adjustment  is  then  multiplied  by  each 
accumulation  in  turn,  and  the  result  is  added  to  or  sub- 
tracted from  the  proper  item.  For  a  detailed  discussion 
with  tables  see  the  Sprague-Perrine  "Accountancy  of 
Investment".    A  last  method  which  may  be  suggested  is 

to  multiply  the  error  by  —  at  rate  i,  and  add  the  result 

Sn 

to  the  Income  items  or  subtract  from  them  as  may  be 

necessary. 

Serial  bonds:  Bonds  are  often  issued  under  an  agree- 
ment that  they  may  be  redeemed  at  various  dates  rather 
than  all  at  once.  These  periodic  redemptions  may  take 
place  semiannually,  annually,  or  at  intervals  of  several 
years.  The  bonds  may  be  redeemable  at  par  or  at  a 
premium  or  discount;  in  practise  they  are  oft^n  redeem- 
able at  a  premium  which  decreases  as  the  final  redemption 
date  approaches.  They  may  be  redeemable  at  the  pleasure 
of  the  trustee  of  the  sinking  fund  or  in  stated  amounts, 
regular  or  otherwise. 

If  the  trustee  is  instructed  merely  to  purchase  bonds 
in  open  market  to  the  amount  of  the  fund  in  his  hands 
at  any  time,  it  is  obviously  impossible  to  value  the  series 


110 


MATHEMATICS  FOR  THE  ACCOUNTANT 


with  any  mathematical  accuracy.  For  such  redemptions 
will  vary  with  varying  market  conditions,  both  price  and 
amount  being  variables. 

If  the  schedule  of  redemptions  is  known,  dates,  amounts 
and  prices  being  stated,  the  series  can  be  valued  as  a 
whole.  No  single  bond  can  ever  be  valued  except  at  a 
so-called  average  price,  which  is  only  an  arbitrary  valua- 
tion, unless  the  redemption  date  of  that  particular  bond 
is  stated.  The  valuation  of  an  irregular  series  proceeds 
as  follows: 

Problem:  A  series  of  $100,000.00  of  bonds  paying  5% 
annually  is  issued  on  January  1,  1920,  redeemable  on 
and  after  January  1,  1926,  in  these  amounts: 

Jan.  1,  1926 $10,000.00 

Jan.  1,  1927 15,000.00 

Jan.  1,  1928 20,000.00 

Jan.  1,  1929 20,000.00 

Jan.  1,  1930 35,000.00 

Value  this  series  on  a  4%  annual  basis. 

It  is  first  necessary  to  schedule  the  interest  payments: 

Jan.  1,  1921,  to  Jan.  1,  1926      $5,000.00  each  year 

Jan.  1,  1927 4,500.00 

Jan.  1,  1928 3,750.00 

Jan.  1,  1929 2,750.00 

Jan.  1,  1930 1,750.00 

The  value  of  the  series  on  a  4  %  annual  basis  is : 
Jan.  1,  1921,  to  Jan.  1,  1925, 

5,000  X  as $22,259.1117 

Jan.  1,  1926,  15,000  X  v«. .  .  .  11,854.7180 
Jan.  1,  1927,  19,500  X  v^. . . .  14,818.3973 
Jan.  1,  1928,  23,750  X  v^. . . .  17,353.8924 
Jan.  1,  1929,  22,750  X  v*. . . .  15,983.8483 
Jan.  1,  1930,  36,750  X  v'".. .  .     24,826.9832 

Total $107,096.8909 


VALUATION  OF  BONDS  111 

If  a  series  is  regular  in  every  way,  redeemable  in  equal 
amounts  at  a  fixed  price  and  at  equal  intervals  of  time,  it 
can  be  valued  by  Makeham's  formula.  Consider  a  series 
of  ten  $1,000.00  bonds,  5%  J  J,  redeemable  one  each  six 
months  beginning  July  1,  1922,  at  par.  This  series  is  to 
be  valued  as  of  Jan.  1,  1917,  ex  interest  on  a  6%  basis. 
The  method  of  stating  the  coupon  rate  is  a  common 
method,  and  means  that  the  bonds  pay  23^%  on  the  first 
of  January  and  July. 

This  is  a  10-period  series  deferred  10  periods.  So  in 
Makeham's  formula  the  present  worth  factor,  instead  of 
v,  is  the  factor  for  a  deferred  annuity,  which  is  an+m 
— SLm,  where  m  is  the  term  of  deferment.  In  this  case  it  is 
a  2  0  — Si  1 0 

C  =  $10,000.00 

K  =  1,000  X  a2o— aio  at  3% 

or  6,347.2720 

C-K  =  3,652.7280 

-^^^  X    3,652.7280         =  3,043.9400 

no  — l , 

$9,391.2120 
The  problem  is  usually  complicated  by  the  fact  that 
redemptions  are  annual,  while  interest  payments  are  semi- 
annual. In  such  a  case  the  yield  is  also  semiannual,  and 
the  best  method  is  to  annualize  both  coupon  and  income 
rates.  A  series  of  five  $10,000.00  bonds,  5%  JJ,  issued 
Jan.  1,  1920,  is  redeemable  at  par,  $10,000.00  at  the  end 
of  each  year  for  five  years.  Value  the  series  on  a  43^% 
basis. 

The  rates  can  be  changed  to  an  annual  effective  basis 
by  the  rule  on  page  35. 


(1.025)2 

1.050625 

annualized  c        = 

5.0625% 

(1.0225)2 

1.0455 

annualized  i         = 

4.55% 

112  MATHEMATICS  FOR  THE  ACCOUNTANT 

Next  we  have  to  find  a  5  at  4.55% 

1.0455                 nl  .01932398745 

times  5  .09661993725 

colog  9.90338006275—10 

In  =  V  °  .80053466 

from  1  .19946534 

divided  by  .0455     =  4.3838536  =  a  n 

Now,  using  Makeham's  formula 

K  =  10,000  X  an  =  $43,838.54 

C-K  =  6161.464 

5.0625 


4.55 


X  6161.464  =  6,855.47 

$50,694.01 


If  the  bonds  are  to  be  redeemed  by  an  amortization 
scheme  the  valuation  is  no  more  difficult.  Consider  a 
series  of  five  $10,000.00  bonds,  paying  5%  annually,  to 
be  amortized  over  a  period  of  ten  years,  the  first  amor- 
tization to  take  place  at  the  end  of  one  year.  To  value  the 
series  on  a  4%  annual  basis. 

—  at  5%  =  .12950548 

aio 

50,000  X  .12950548       =       $6,475,229 

This  is  the  annual  amortization  payment,  disregarding 
the  necessity  of  redeeming  whole  bonds.  If  this  annuity 
is  purchased  on  a  4%  annual  basis, 

aioat4%  =  8.11089578 

times  6.475.229  =     $52,519.9075 

Redemption  of  a  series:  As  has  been  said,  a  series  may 
be  redeemed  at  the  discretion  of  the  trustee,  or  the 
arrangements  as  to  amounts  and  dates  may  be  stated  in 


VALUATION  OF  BONDS  113 

the  deed  of  trust  under  which  the  bonds  were  issued. 
Two  methods  which  lead  to  some  mathematical  diffi- 
culties will  now  be  discussed. 

It  may  be  desirable  to  retire  the  series  by  an  amorti- 
zation scheme,  so  that  periodic  payments  on  account  of 
principal  and  coupons  combined  may  be  as  nearly  equal 
as  possible.  Of  course  the  amount  of  principal  repaid  at 
each  redemption  date  must  be  an  exact  multiple  of  the 
face  of  the  bonds.  In  such  a  case  the  periodic  payments 
cannot  be  exactly  equal,  and  the  schedule  requires  con- 
siderable adjustment.  Also  this  situation  is  not  to  be 
confused  with  the  sinking  fund  method,  by  which  no 
bonds  are  redeemed  until  the  entire  series  matures,  and 
periodic  payments  are  allowed  to  accumulate  interest  in 
the  hands  of  the  trustee. 

Suppose  an  issue  of  50  bonds  of  par  SI, 000.00  is  to  be 
retired  over  a  period  of  ten  years,  beginning  one  year 
from  date.  Interest  on  bonds  outstanding  is  to  be  at  the 
rate  of  5%  annually.  The  amortization  factor  is 
.12950458,  which,  multipHed  by  50,000,  gives  the  average 
annual  cost  $6,475,229,  The  first  year's  coupons  are 
$2,500.00,  leaving  $3,975,229  for  redemptions.  This  will 
almost  redeem  four  bonds,  and  it  will  be  good  policy  to 
appropriate  the  necessary  $24.77.  The  annual  payment 
then  is  $6,500.00,  and  the  par  of  bonds  outstanding  at  the 
beginning  of  the  second  year  is  $46,000.00.  5%  interest 
for  the  second  year  is  $2,300.00,  leaving  $4,175,229  for 
redemptions.  It  will  be  best  to  appropriate  only  $4,000.00. 
The  extra  $175,229  may  be  appropriated  if  desired  and 
left  to  draw  interest  and  later  it  can  be  used  to  make  up 
deficiencies  such  as  occurred  in  the  first  year.  This  is 
ultra  conservative,  however.  The  schedule  can  now  be 
given  without  comment: 


114  MATHEMATICS  FOR  THE  ACCOUNTANT 

Schedule  of  Amortization  of  $50,000.00  Bonds  at  Par 

$1,000.00,   with  Interest  at  5%  Annually,   in 

Ten  Equal  Annual  Instalments 

Par  of  Number 

Bonds  Out-  Interest      Total  Net      of  Bonds 

Year    standing         5%      Payment  Payment  Retired 

1  $50,000.00  $2,500.00  $6,500.00  $4,000.00        4 

2  46,000.00  2,300.00  6,300.00  4,000.00  4 

3  42,000.00  2,100.00  6,100.00  4,000.00  4 

4  38,000.00  1,900.00  6,900.00  5,000.00  5 
6  33,000.00  1,650.00  6,650.00  5,000.00  5 

6  28,000.00  1,400.00  6,400.00  5,000.00  5 

7  23,000.00  1,150.00  6,150.00  5,000.00  5 

8  18,000.00  900.00  6,900.00  6,000.00  6 

9  12,000.00  600.00  6,600.00  6,000.00  6 
10  6,000.00  300.00  6,300.00  6,000.00  _6 

14,800.00  64,800.00  50,000.00  50 
The  usual  checks  should  be  observed.  If  there  were  a 
question  regarding  the  payment  on  account  of  redemp- 
tions in  any  year,  it  would  be  wise  to  redeem  too  few 
bonds,  rather  than  too  many.  This  satisfies  the  bond- 
holders and  allows  the  issuing  company  the  use  of  the 
money  for  one  year  more.  In  general  the  number  of 
bonds  redeemed  is  progressive,  increasing  slowly  from 
the  first  date  to  the  last. 

The  other  method  which  we  ought  to  consider  is  the 
method  which  requires  what  we  may  term  a  redemption 
fund.  By  this  method  annual  costs  are  equalized  regard- 
less of  amount  of  redemptions.  This  is  done  by  appro- 
priating a  fixed  amount  and  allowing  any  excess  to  remain 
in  the  bank  until  it  is  needed.  In  this  way  interest  is 
earned  on  all  unused  balances,  yet  they  are  available 
without  any  strain  on  the  finances  of  the  issuing  com- 
pany. 


VALUATION  OF  BONDS  115 

A  series  of  $100,000.00  of  5%  bonds  such  as  we  valued  on 
page  110  is  a  good  example.  Suppose  we  appoint  a  trustee 
and  agree  to  pay  him  equal  amounts  at  intervals  of  six 
months  from  six  months  after  the  date  of  issue  to  the 
date  of  the  last  redemption.  This  arrangement  does  away 
with  the  necessity  of  annualizing  rates  an^  simplifies  the 
problem.  The  trustee  is  to  meet  all  coupons  and  redemp- 
tions and  keep  the  balance  of  the  fund  invested  at  a  rate 
which  we  assume  will  be  not  less  than  4%  annually. 
The  balance  thus  invested  must  be  available  in  any  year 
when  total  redemptions  and  coupons  exceed  the  annual 
payment  into  the  fund. 

First  it  is  necessary  to  value  the  series  at  the  date  of 
issue.  This  value  has  already  been  found  to  be 
$107,096.8909.    The  annual  payment  to  meet  this  must 

be  107,096.9809  times  — at  4%,  which  is  .12329094;  the 

aio 

product   is    the   annual    payment,    $13,204.0763.     The 
schedule  is  given  on  the  following  page. 

Instalment  bonds:  Sometimes  a  bond  is  payable, 
principal  and  interest,  at  regular  intervals  and  in  equal 
amounts.  This  is  the  usual  amortization  problem.  Con- 
sider a  bond  for  $5,000.00  to  be  repaid  in  ten  semiannual 
instalments  with  interest  on  outstandings  at  4%  nominal, 
convertible  semiannually,  payments  to  be  equal  in 
amount.  The  bond  is  to  be  priced  on  a  3%  basis.  The 
periodic  payment  is  5,000  X  .11132653  =  $556.6327. 
The  present  worth  of  these  ten  payments  at  1^%  is 
556.6327  X  9.22218455  =  $5133.3695. 
The  two  operations  may  be  expressed  in  one  formula: 

A  =  C  (  —  at  rate  c)  X  (an  at  rate  i). 
an 


116  MATHEMATICS  FOR  THE  ACCOUNTANT 

Determining  the  income  rate  on  a  bond:  This  is  a  sub- 
ject into  which  the  accountant  need  not  go  as  deeply  as 
the  actuarial  writers,  but  a  fe^;  of  the  simpler  methods 
may  be  useful  at  times. 

For  most  purposes  the  income  rate  can  be  read  accur- 
ately enough  from  bond  tables.  If  the  rate  obtained  from 
the  tables  is  not  acceptable,  a  closer  rate  may  be  obtained 
by  interpolation  between  two  table  values.  Selecting  the 
nearest  values  above  and  below  the  given  value  proceed 
as  in  finding  logarithms.  A  $1,000.00  bond  redeemable 
at  par  after  40  years,  bearing  interest  at  4%,  is  bought  at 
93.50,  that  is,  $935.00.  In  the  tables  for  a  40-year  4% 
bond  we  find 

yield        4.35  cost        $942,955 

4.30  952.116 


differences 

.05 

9.161 

partial  differences 

X 

2.116 

x:  .05 

=  2.116:9.101 

X 

=    .1155 

rate 

=  4.26155% 

VALUATION  OF  BONDS 


117 


d 

O 


fl    S    c o  o 

p    a;    O O  O 

§  'S  '^    :    :    :    :    '  '=''^^ 


lis  MATHEMATICS  FOR  THE  ACCOUNTANT 

After  finding  table  values  it  is  possible  to  proceed  by 
the  method  of  trial  and  error.  That  is,  in  the  bond  just 
discussed,  after  finding  the  4.26155%,  and  ascertaining 
that  the  corresponding  price  is  $949.99,  the  yield  rate  can 
be  slightly  increased  and  the  bond  valued  logarithmically. 
Of  course  this  will  eventually  give  as  correct  a  value  as 
the  number  of  places  used  can  offer.  It  is  veiy  laborious, 
and  generally  unnecessary.  It  should  be  borne  in  mind 
that  too  high  a  yield  rate  results  in  too  low  a  price;  and 
too  low  a  yield  rate  results  in  too  high  a  price. 

The  next  method  is  given  by  Sprague,  and  is  perhaps  the 
best  of  all,  in  that  the  result  of  each  step  checks  up  the 
assumed  rate.  It  is  based  on  a  $100.00  bond,  and  may  be 
expressed  in  the  following  rule: 

1)  Assume  an  income  rate;  half  of  this  will  be  i; 

2)  find  an  for  $1.00  at  rate  i; 

3)  take  hdf  the  result  of  2) ; 

4)  divide  the  result  of  3)  into  the  known  premium 
or  discount; 

5)  if  the  bond  was  bought  at  a  premium,  subtract 
the  result  of  4)  from  c ;  if  bought  at  a  discount,  add ; 

6)  compare  the  result  of  5)  wuth  1);  modify  1) 
and  repeat  until  the  result  of  5)  is  near  enough 
to  1)  to  be  acceptable. 

Problem :  To  determine  the  income  rate  on  a  5  %  bond 
bought  at  110,  redeemable  at  par  after  20  years.  The 
successive  steps  may  be: 

Trial  Resulting 

Rate  a  n  Rate 

4  27.35548         4.268885 

4.1  27.11714    4.262459 

4.2  26.88179    4.25341 
4.25      26.76526    4.252763 

I     4.252     26.76254    4.252687 
4.2524    26.75967    4.25261 


VALUATION  OF  BONDS  119 

The  income  rate  is  apparently  about  4.2525.  Valuing  the 
bond  at  that  rate  we  find  $1,100.0134. 

Two  of  the  many  methods  given  by  Todhunter  are 
simple  and  fairly  accurate  if  used  with  care. 

First,  Let  k  be  the  premium  per  dollar  of  redemption 

A— C- 
value  ;:that  is  k  =  — — — '  then 

k 


1  = 


'-^x*^ 


Problem:  Determine  the  income  rate  on  a  4^%  bond 
redeemable  at  $1,125.00  after  25  years,  bought  at  120. 
Adjusting  c,  the  periodic  coupon  rate  is  2%. 

A  =  $1,200.00 

C  =     1,125.00 

A-C  =         75.00 

divided  by  C         =  .06666 this  is  k 

divided  by  50        =  .0013333 

from  c  .0186666 

This  is  the  numerator  of  the  formula. 

n  +  1  =51 

2n  =       100 

quotient  .51 

times  k  .034 

add  1  1.036 

This  is  the  denominator. 

The  quotient  is  1.8053  semiannually, 

or  3.6106%  nominal. 

The  correct  rate  is  3.5936%,  an  error  of  .017  annually. 
If  the  bond  is  bought  at  a  premium  this  result  will  be  too 
high;  if  bought  at  a  discount  the  result  will  be  too  low. 


120 


MATHEMATICS  FOR  THE  ACCOUNTANT 


Second:  Assume  a  rate  i'  and  compute  K'  as  for  Make- 
ham's  formula.  If  K'  is  not  near  enough  to  the  given  A, 
modify  the  rate  i'.  This  part  of  the  solution  cnn  be 
shortened  by  use  of  bond  tables.  After  a  satisfactory  K' 
has  been  found,  use  this  formula: 
"C-A 


i  =  c  +  i' 


Lc-K'J 


For  the  bond  just  considered  the  solution  is: 

Assume  i'  =  1.8%  semiannually. 

To  find  K'  at  this  rate: 


yn       = 


1125 


.007747778 
.3873889 
9.6126111—10 
.409837 
461.0666     this  is  K' 
-1200    1 


1.018  nl 

times  50 

colog 

In 

times  1,125 

i  =  2  +  1.8, 

L1125-461.0666J 

Note  that  the  numerator  of  the  parenthesis  is  negative, 
so  that  the  value  of  the  fraction  must  be  subtracted 
from  2. 

i  =  1.796664% 

or  3.5933%,  annually. 


PROBLEMS  ON  CHAPTER  XI 

Value  these  bonds  (rates  c  and  i  are  quoted  for  the 
whole  year) : 


122.  par  $1,000.00  c 

'  4% 

i3% 

n  40  years 

123.            1,000.00 

33^% 

23^% 

5  build  a  schedule 

124.            1,300.00 

6% 

3.6% 

15 

125.            1,000.00 

4% 

5% 

45 

126.            1,000.00 

4>^% 

33^% 

333^ 

127.            1,000.00 

4% 

3% 

12 

VALUATION  OF  BONDS  121 

128.  1,000.00      7%        5%  10 

redeemable  at  108 

129.  1,000.00      3H%    43^%      20 

redeemable  at  105 

130.  1,000.00      5%JJ   3%  due  Jan.  1,  1918; 

purchased  April  1,  1915 

131.  1,000.00      7%        5%  interest    June    15 

and  Dec.  15,  due  June  15, 
1916,  purchased  Oct.  1, 
1898 

132.  1,000.00      5%        6%  20  income  tax  15% 

133.  1,000.00      43^%     5%  30  redeemable  at 

102;  income  tax  25% 

134.  1,000.00      33^%    3%  20  income  tax  5% 

135.  A  4%  bond  AO,  redeemable  at  $1,000.00  on  Oct.  1, 
1916,  is  purchased  April  1,  1914,  on  a  6%  basis. 
Value  the  bond  and  set  up  a  schedule  to  be  used  as 
the  source  of  entries  on  a  set  of  books  which  are 
closed  June  30  and  December  31. 

136.  Find  the  value  on  a  4%  basis  of  a  series  of  bonds 
amounting  to  $100,000.00  issued  on  January  1, 
1885,  paying  5%  J  J,  redeemable  $2,000.00  each  six 
months  at  par  on  and  after  July  1,  1910. 

137.  A  series  of  bonds  amounting  to  $50,000.00,  bearing 
53^%  annually  is  to  be  redeemed 

$8,000.00  at  the  end  of  one  year  at  105, 
9,000.00  at  the  end  of  two  years  at  103, 
10,000.00  at  the  end  of  three  years  at  102, 
11,000.00  at  the  end  of  four  years  at  101, 
12,000.00  at  the  end  of  five  years  at  100. 
Value  the  series  on  a  43^%  annual  basis. 

138.  A  series  of  bonds  to  the  value  of  $250,000.00  bearing 
5%  annually  is  to  be  amortized  over  a  period  of  ten 
years.    Value  the  series  on  a  6%  annual  basis. 


122  MATHEMATICS  FOR  THE  ACCOUNTANT 

139.  Set  up  a  schedule  for  the  company  issuing  the  above 
bonds  showing  how  the  annual  payments  on  account 
of  coupons  and  redemptions  combined  may  be  kept 
as  nearly  equal  as  possible.  Redemptions  will  be 
at  par  S1,000.00. 

140.  A  series  of  bonds  to  the  amount  of  $500,000.00, 
bearing  6%  annually,  is  issued  January  1,  1908. 
Par  of  the  bonds  is  $1,000.00.  For  10  years  they  pay 
interest  only,  and  beginning  January  1,  1919,  the 
series  is  to  be  amortized  over  a  period  of  20  years. 
What  must  be  the  annual  payment  into  a  redemption 
fund  which  can  be  accumulated  at  43^2%  annually 
for  the  full  thirty  years  from  January  1,  1908,  to 
January  1,  1938,  to  meet  the  interest  charges  and 
retire  the  bonds? 

141.  A  series  of  bonds  to  the  value  of  $50,000.00  is  issued 
under  an  agreement  that  they  shall  be  redeemed 
$10,000.00  each  year  for  five  years.  They  pay 
6%  JJ,  and  are  purchased  on  a  5%  annual  basis. 
What  is  the  price? 

Find  the  income  rate  on  the  following  bonds;  if  possible 
use  the  method  given  by  Sprague  (seepage  118),  and  check 
by  one  of  the  methods  taken  from  Todhunter,  or  else 
by  valuing  the  bonds  logarithmically.  Bond  tables  may 
be  used  for  first  approximations,  but  answers  must  be 
accurate  to  hundredths  of  a  percent. 

142.  A  5%  50-year  bond,  redeemable  at  par,  bought  at 
110. 

143.  A  4%  20-year  bond,  costing  115,  redeemable  at  108. 

144.  A  5%  20-year  bond  is  quoted  at  "95  to  yield  about 
6.4%."  Is  this  accurate?  If  not,  quote  a  more 
accurate  yield  rate. 


CHAPTER  XII 
THE  SLIDE  RULE 

This  chapter  is  a  brief  explanation  of  only  such  uses 
of  the  slide  rule  as  are  useful  in  general  accounting 
practise.  The  real  study  of  the  slide  rule  must  be  in  the 
form  of  practise;  simple  problems  should  be  solved  by 
arithmetic  and  the  solutions  checked  by  the  slide  rule. 
Only  in  this  way  can  proficiency  be  attained. 

The  slide  rule  is  a  mechanical  device  for  performing 
multiplication,  division,  raising  to  a  power  (by  continued 
multiplication),  square  roots,  and  in  general  the  easier 
operations  which  can  be  performed  by  the  use  of  logar- 
ithms. 

Among  the  commercial  operations  for  which  the  slide 
rule  is  used  are  payrolls,  invoices,  discounts,  setting  prices 
on  goods  to  be  sold,  inventories,  determining  rates  of 
profit  and  loss,  proportion,  foreign  prices  and  exchange, 
simple  interest  and  discount. 

The  mechanical  and  engineering  uses  are  numerous, 
and  valuable  to  the  cost  accountant.  Such  processes 
should  be  taught  in  engineering  courses,  however,  and 
will  not  be  mentioned  here. 

There  are  many  sizes  and  forms  of  calculating  machines 
built  on  the  logarithmic  principle.  Beside  the  usual 
eight  and  ten-inch  slide  rules  there  are  twenty-inch  rules 
for  desk  use.  Special  rules  have  trigonometric  scales, 
scales  of  equal  parts  for  use  with  logarithmic  tables,  cube 
root  scales,  and  so  on.  Other  machines  are  circular  or 
cylindrical,  thus  securing  greater  accuracy  by  reason  of 
their  longer  scales. 

Our  discussion  will  concern  the  shorter  slide  rules, 
which  are  most  useful  to  the  accountant.    Computations 

123 


124  MATHEMATICS  FOR  THE  ACCOUNTANT 

with  the  ten-inch  rule  are  accurate  to  the  third  digit, 
and  the  fourth  place  can  be  estimated  with  a  fair  degree 
of  accuracy  when  the  result  is  at  the  left  end  of  the 
scales.  Results  to  more  than  four  places  are  useful  only 
to  check  actual  computation. 

The  logarithmic  scale:  The  logarithms  of  the  numbers 
from  1  to  10  are: 

1  nl      0 

2  .301030 

3  .477121 

4  .602060 

5  .698970 

6  .778151 

7  .845098 

8  .903090 

9  .954243 
10  1 

If  we  multiply  each  of  these  logarithms  by  ten  and  lay 
the  lengths  off  in  sequence  on  a  line  ten  inches  long  we 
have  this  scale: 

wmm 


The  usefulness  of  such  a  scale  consists  in  the  fact  that 
as  multiplication  can  be  performed  by  adding  logarithms, 
so  multiplication  can  be  performed  by  adding  lines  whose 
logarithmic  measures  correspond  to  the  given  numbers. 
For  instance,  the  sum  of  the  logarithms  of  2  and  3  is 
.778151,  which  is  the  logarithm  of  6.  If  we  have  two 
lines  whose  lengths  are  respectively  the  logarithms  of 
2  and  3,  the  sum  of  the  lengths  must  be  the  logarithm 
of  6.    This  statement  can  be  tested  from  the  above  scale 


THE  SLIDE  RULE  125 

with  the  edge  of  a  sheet  of  paper  or  any  other  straight 
edge,  whether  it  has  any  scale  marked  on  it  or  not. 

A  slide  rule  is  composed  of  three  parts:  stock,  slide, 
and  runner.  The  runner  has  a  line  or  wire  on  its  glass, 
to  assist  in  locating  or  keeping  the  location  of  points. 
On  the  face  of  the  rule  are  four  scales,  lettered  A,  B,  C 
and  D ;  A  and  D  are  on  the  stock,  B  and  C  are  on  the  slide. 
A  and  B  are  alike,  being  scaled  from  1  to  10  twice;  C 
and  D  are  scaled  only  once.  In  other  words,  A  and  B 
are  "double  scales". 

Consider  first  the  D  scale;  it  is  precisely  like  the  scale 
just  constructed  from  the  logarithms,  with  the  addition 
of  subdivisions.  These  subdivisions  are  not  on  a  logar- 
ithmic scale,  but  are  equal  divisions  of  the  section  of  the 
scale  in  which  they  occur.  This  corresponds  to  the 
manner  in  which  we  interpolate  in  the  use  of  logarithms, 
by  ordinary  proportion.  Of  course  such  results  are  not 
strictly  accurate,  but  they  are  satisfactory  for  ordinary 
use.  The  meaning  of  these  subdivisions  is  evident.  If 
we  suppose  that  the  1  at  the  left  is  10,  then  the  subdivision 
next  it  marked  1  is  11 ;  and  the  subdivision  next  after  that 
is  12.  In  the  division  beginning  with  2,  the  subdivision 
marked  1  is  21,  and  so  on.  The  subdivision  which  we 
called  11  is  marked  off  into  ten  very  small  parts;  the  first 
of  these  is  111  with  the  decimal  point  wherever  the 
problem  requires;  and  so  on.  If  the  1  at  the  extreme  left 
were  1000,  then  the  subdivision  1  would  be  1100,  and  the 
sub-subdivision  1  would  be  1110. 

The  scale  begins  at  1  because  the  logarithm  of  1  is  0, 
and  zero  length  is  at  the  beginning  of  the  scale.  This 
1  may  be  1  or  100  or  1,000,000  or  .01;  the  logarithms  of 
all  these  numbers  differ  only  in  their  characteristics, 
and  the  slide  rule  is  a  scale  of  mantissae  and  not  of 
characteristics.     The   characteristic   in   any   result   can 


126  MATHEMATICS  FOR  THE  ACCOUNTANT 

usually  be  determined  by  common  sense  although  there 
are  rules  which  will  be  given  later. 

The  A  scale  differs  from  the  D  scale  only  in  that  the 
logarithmic  scale  is  repeated  and  the  subdivisions  are 
less  numerous  and  shorter.  If  the  1  at  the  extreme  left 
is  1,  then  the  1  in  the  middle  is  10;  if  the  1  at  the  left  is 
100,  the  1  in  the  middle  is  1000.  If  the  1  at  the  left  is 
.01,  the  1  in  the  middle  is  .10;  and  so  forth.  Any  number 
on  the  A  scale  is  the  square  of  the  number  directly  below 
it  on  the  D  scale.  For  instance,  move  the  runner  until 
the  wire  is  over  9  (not  90)  on  the  A  scale;  directly  under 
the  wire  on  the  D  scale  find  3. 

The  rules  for  determining  the  characteristic  are : 
In  multiplication: 

If  the  slide  projects  to  the  right,  the  characteristic  is 
the  sum  of  the  characteristics  of  the  factors  plus  1; 

If  the  slide  projects  to  the  left,  sum  of  the  character- 
istics. 

In  division: 

If  the  slide  projects  to  the  right,  the  characteristic  is 
the  characteristic  of  the  dividend  less  that  of  the 
divisor; 
If  the  sHde  projects  to  the  left,  this  difference  less  1. 
A  few  simple  illustrations : 
Multiplication:  3  X  4  =  12. 

On  C  and  D;  set  1  of  C  to  3  of  D;  below  4  on  C  read 
12  on  D.  Notice  that  in  order  to  get  4  of  C  over  the 
D  scale  at  all,  the  slide  must  be  moved  to  the  left. 
The  characteristic  is  determined  by  the  first  clause 
of  the  rule  for  multiplication. 
On  A  and  B;  set  1  of  B  to  3  (not  30)  of  A;  on  A  over 
4  of  B  read  12.    Note  that  the  slide  projects  to  the 


THE  SLIDE  RULE  127 

right,  but  the  characteristic  is  1  as  before;  this  is 
because  the  result  is  in  the  right  hand  half  of  the 
double  scale  A. 

Division:  35  ^5  =  7. 

On  C  and  D;  set  the  runner  to  35  of  D;  bring  5  of  C 
to  the  runner;  below  1  of  C  read  7  on  D.  The 
difference  of  the  characteristics  is  1;  but  the  slide 
projects  to  the  left,  so  the  characteristic  of  the 
quotient  is  1  less  than  that  difference.  See  the  last 
part  of  the  rule  for  the  characteristic  in  division. 

On  A  and  B:  set  the  5  of  B  to  35  of  A;  over  1  of  B 
read  7  on  A. 

Notice  that  the  divisor  is  on  the  slide,  and  is  set 
opposite  the  dividend  on  the  stock. 

To  square  a  number:  7'  =  49. 

Set  the  runner  to  7  of  D;  on  A  under  the  runner  read 
49. 

Square  root:  to  find  the  square  root  of  81. 

Set  the  runner  to  81  of  A;  under  the  runner  on  D 
read  9. 
Proportion:  Most  engineering  problems  involve  the  use 
of  "gauge  points",  which  are  ratios  comparing  two  quan- 
tities; for  instance,  the  diameter  and  circumference  of  a 
circle;  the  speed  of  two  shafts;  and  such  ratios.  Com- 
mercial gauge  points  are  the  ratio  of  a  yard  to  a  metre, 
foreign  exchange  rates,  and  many  others.  There  is  always 
a  list  of  such  gauge  points  as  are-  constant  on  the  back 
of  a  slide  inile.  A  few  illustrations: 
What  number  of  yards  is  equivalent  to  350  metres?  The 
gauge  point  is  75 :82 ;  that  is,  75  metres  are  equivalent  to 
82  yards.  Set  75  of  C  to  82  of  D;  under  350  of  C  read 
382  on  D. 


128  MATHEMATICS  FOR  THE  ACCOUNTANT 

What  is  the  value  in  francs  of  $125.00,  exchange  at  5.26. 
Proceed  as  in  multipHcation :  125  X  5.26  =  657.5. 
At  the  same  quotation  what  is  the  value  in  dollars  of 
2,000  francs?  This  is  division.  Set  5.26  of  C  to  2,000 
of  D;  under  1  of  C  read  3803.  The  characteristic  is 
3—0  —  1  =  2;  answer  $380.30.  Correct  value,  $380.25. 
An  article  costing  $2.75  is  to  be  sold  to  make  75%  on  the 
cost.    Find  the  seUing  price..  . 

The  selling  price  is  175%  of  the  cost.  Multiply  2.75  by 
1.75,  product  $4.82. 

An  article  costing  $3.25  is  to  be  sold  to  gain  55%  on  the 
selling  price.  Find  the  selling  price.  Evidently  the  cost 
is  45%  of  the  selling  price,  and  the  solution  requires 
division.  Set  .45  of  C  to  3.25  of  D;  under  1  of  C  read 

$7.22. 

Sometimes  this  use  of  gauge  points  leads  to  a  puzzling 
situation. 

Problem:  Find  the  weight  of  10  gallons  of  water.  The 
gauge  point  is  3:25;  that  is,  3  gallons  weigh  25  pounds. 
Set  3  of  C  to  25  of  D;  the  answer  is  expected  under  105 
of  C,  but  this  point  is  outside  the  stock.  The  shde  must 
be  thrown  to  the  right  its  whole  length ;  to  assure  accuracy, 
set  the  runner  to  the  one  at  the  right  end  of  the  slide,  and 
then  throw  the  shde  over  until  the  one  at  its  left  end  is 
under  the  runner.  This  multiplies  the  result  by  10 
Under  105  of  C  read  87.5. 

Successive  operations:  What  is  the  interest  on  a  $50.00 
Liberty  Bond  at  3H%  from  November  15  to  May  16? 
The  time  is  181  days.  Expressed  as  cancellation,  the 
problem  is: 

50  X  .035  X  181 
365 


THE  SLIDE  RULE  129 

Set  1  of  C  to  50  of  D ;  runner  to  .035  of  C,  and  under  it 
is  one  year's  interest  (characteristic  0).  Set  1  of  C  to 
this  result,  runner  to  181  of  C  (characteristic  2).  Set 
365  of  C  to  the  runner,  and  under  1  of  C  read  87.  (Charac- 
teristic  2-2  —  1  =  —1).    Answer  .87. 


PROBLEMS  ON  CHAPTER  XII 
A.  Drill  problems  to  train  in  fundamentals: 


36  X  14 

82  X  74 

875  -^35 

4588  H- 

74 

28  X  36  ^ 

63 

mxl 

23 « 

23' 

>J/1681 

^15376 

23* 


B.  Commercial  problems : 

145.  What  amount  is  due  to  an  employee  who  has  worked 
46  M  hours  at  $18.00  for  a  48-hour  week? 

146.  What  amount  is  due  this  employee  if  he  works 
334  hours'  overtime,  at  "time  and  a  half?" 

147.  A  chair  is  listed  at  $16.50  less  10—10-^5%.  What 
is  the  net  cost? 

148.  Name  a  selling  price  on  the  above  chair  to  gain 
75  %  on  the  cost. 

149.  A  60-day  note  for  $318.00  is  dated  May  9  with 
interest  at  53^%  exact  time.  What  is  the  value  at 
maturity? 

150.  The  above  note  is  discounted  May  11  at  6%,  bankers' 
time.    Find  the  net  proceeds. 

151.  What  is  the  proceeds  of  a  Russian  bill  for  550  rubles, 
exchange  at  .12^? 


130  MATHEMATICS  FOR  THE  ACCOUNTANT 

152.  Wliat  is  the  proceeds  of  a  draft  on  London  for 
£385  43.,  exchange  at  4.855? 

153.  What  draft  on  London  can  be  purchased  for 
$1,250.00,  exchange  at  4.8675? 

154.  When  gold  is  quoted  at  $20.60  per  ounce,  what  is 
the  price  in  francs  per  gramme,  exchange  at  5.75? 
Gauge  point,  ounces  to  grammes  6:170. 

155.  What  is  the  adjusted  coupon  rate  on  a  33^%  bond 
redeemable  at  108? 

156.  In  taking  an  inventory  in  a  hardware  store  38 
clamps  were  counted.  They  were  billed  from  the 
manufacturer  at  $6.50  per  gross  less  25 — 10—10%. 
The  dealer  priced  them  to  gain  50%  on  net  cost. 
What  is  the  inventory  value,  depreciation  excluded? 
(Value  one  clamp  and  multiply  by  38.) 

157.  A  desk  costing  $85.00  is  to  be  priced  to  gain  40% 
on  the  selling ^price.    State  the  selling  price. 

158.  A  sprinter  rah  100  metres  in  11  seconds.  What  is 
the  equivalent  speed  for  100  yards? 

159.  An  article  costing  $25.00  is  sold  at  $40.00  less 
10 — 10%.    What  is  the  percent  of  profit  on  cost? 

160.  The  financial  report  of  a  business  showed  the  follow- 
ing facts,  in  thousands  of  dollars: 

Sales 315 

Cost  of  Sales 189 

Selling  Expenses 68 

General  Expenses 22 

Profit ? 

What  percent  of  Sales  is  each  of  the  four  items? 
Total  must  be  approximately  100%. 


CHAPTER  XIII 
REVIEW  PROBLEMS 

161.  What  is  the  cash  balance  October  1,  at  5%  exact 
time,  of  the  following  account: 

Debits:      May    8,    $500.00,    2    months;    June    25, 
$1,000.00,  60  days;  August  10,  $500.00,  cash. 
Credits:    July  1,  $1,200.00,  30  days;  September  15, 
$200.00,  cash. 

162.  When  a  boy  was  born  $500.00  was  placed  to  his 
credit  in  a  bank  which  pays  5%  nominal,  com- 
pounding semiannually.  If  the  account  is  not  dis- 
turbed until  his  twenty-first  birthday,  what  will 
be  the  balance  on  that  date? 

163.  A  man  buys  a  farm  for  $10,000.00,  paying  cash 
$2,500.00  and  agreeing  to  pay  the  balance  with 
accrued  interest  in  three  equal  annual  instalments, 
interest  at  6%  annually.  What  is  the  annual  pay- 
ment?   Prove  by  a  schedule. 

164.  A  clerk  expects  to  go  into  business  for  himself  as 
soon  as  he  has  saved  $5,000.00.  If  he  has  now 
$2,500.00,  and  can  invest  all  funds  at  4)^%  annually, 
how  much  must  he  save  yearly  so  that  he  may  have 
the  necessary  amount  at  the  end  of  five  years? 

165.  How  many  years  will  be  necessary  to  accumulate 
$1,000.00  by  investing  ^150.00  per  annum  at  6% 
annually']?    Solve  without  schedule. 

166.  A  city  issues  bonds  for  $500,000.00  on  which  it  pays 
interest  at  5%  annually.  It  accumulates  a  sinking 
fund  at  4%  annually  to  retire  the  bonds  at  the  end 
of  fifteen  years.    Find  the  annual  payment  into  the 

131 


132  MATHEMATICS  FOR  THE  ACCOUNTANT 

sinking  fund  and  the  amount  in  the  fund  at  the  end 
of  five  years,  without  schedule. 

167.  What  sum  must  be  put  semiannually  into  an  invest- 
ment which  pays  4%  effective  in  order  to  accumulate 
$1,000.00  at  the  end  of  five  years? 

168.  A  person  invests  $5,000.00  at  5%  annually.  Prin- 
cipal and  interest  are  to  provide  a  fixed  income  for 
ten  years,  at  the  end  of  which  time  the  capital  will 
be  exhausted.  The  first  payment  is  to  be  made  one 
year  from  date.  What  is  the  annuity?  Set  up  a 
schedule. 

169.  A  mortgage  for  $5,000.00  was  given  on  January  1, 
1906,  to  be  repaid  in  ten  equal  semiannual  instal- 
ments with  interest,  beginning  July  1,  1906.  Find 
the  periodic  payment  and  set  up  a  schedule,  basing 
on  6%  nominal,  convertible  semiannually. 

170.  According  to  the  conditions  of  a  will  the  sum  of 
$20,000.00  is  to  be  held  in  trust  until  it  has  increased 
to  $25,000.00.  If  the  fund  can  be  invested  at  5% 
nominal,  compounding  semiannually,  when  will  the 
beneficiary  receive  the  money.  Solve  without  schedule. 

171.  A  father  bequeaths  his  son  on  his  tenth  birthday 
$10,000.00  worth  of  preferred  stock  which  pays 
6%  dividends  semi-annually.  The  will  further  directs 
that  the  income  shall  be  invested  by  the  trustee 
until  the  son  is  twenty-one  years  of  age.  Assuming 
that  the  stock  continues  to  pay  dividends  regularly, 
and  that  these  dividends  can  be  invested  at  5% 
nominal,  convertible  quarterly,  find  the  value  of  the 
property  at  the  end  of  the  trustee's  term. 

172.  How  large  an  endowment  is  necessary  for  a  room  in 
a  hospital,  costing  $5,000.00,  to  instal  and  $1,500.00 
annually  for  maintenance?    Assume  4%  annually. 


REVIEW  PROBLEMS  133 

173.  Is  it  more  profitable  for  a  city  to  pay  $3.00  per 
square  yard  for  paving  that  will  last  5  years  or  $4.00 
for  paving  that  will  last  7  years?  Assume  5% 
annually. 

174.  A  business  man  wishes  to  set  aside  on  his  forty-fifth 
birthday  a  sum  which  will  give  him  an  income  of 
$1,000.00  a  year  for  ten  years,  the  first  payment 
to  be  made  on  his  sixtieth  birthday.  On  a  33^% 
annual  basis,  what  is  the  sum  necessary? 

175.  An  issue  of  $50,000.00  of  bonds  of  par  value'$  1,000. 00 
bearing  interest  at  5%  annually,  is  to  be  retired  as 
follows:  For  the  first  five  years  interest  only  will 
be  paid,  after  which  time  the  bonds  and  interest 
will  be  amortized  over  a  period  of  five  years.  The 
money  to  provide  for  all  this  is  to  be  raised  by  equal 
payments  into  a  fund  through  the  full  ten  years; 
the  fund  can  be  accumulated  at  4%  annually.  What 
is  the  amount  of  this  annual  payment?  Construct 
a  schedule. 

176.  A  bond  redeemable  at  par  $1,000.00  on  January  1, 
1920,  bearing  6%  JJ,  is  bought  January  1,  1914,  to 
yield  4%.     Find  the  price? 

177.  A  6%  bond,  JJ,  redeemable  at  par  $1,000.00  on 
January  1,  1919,  is  purchased  July  1,  1916,  to  yield 
10%.    Find  the  price. 

178.  A  manufacturing  concern  contracts  for  a  factory 
site  for  $20,000.00  cash  and  $5,000.00  at  the  end  of 
each  year  for  five  years.  If  money  is  worth  5% 
annually,  what  would  be  a  fair  cash  price  for  the  lease? 

179.  A  man  borrows  $5,000.00,  agreeing  to  pay  6% 
interest  annually,  and  repay  the  principal  at  the 
end  of  ten  years.  He  accumulates  a  sinking  fund 
at  4%  annually.  Find  the  annual  cost  of  carrying 
the  debt. 


134  MATHEMATICS  FOR  THE  ACCOUNTANT 

180.  Would  it  be  better  to  amortize  the  above  debt  over 
the  ten  years  by  semiannual  payments  at  5%  nom- 
inal, semiannually? 

181.  On  his  fortieth  birthday  a  man  begins  to  invest 
$1,000.00  a  year.  On  his  seventieth  birthday  he 
makes  the  usual  payment  and  retires  from  business. 

(a)  If  money  has  been  worth  4%  annually  through- 
out the  period,  what  is  the  value  of  his  investment? 

(b)  Assuming  that  he  will  live  not  more  than  fifteen 
years,  he  plans  to  exhaust  the  investment  in  that 
time.  If  money  will  continue  to  earn  4%,  what  sum 
can  he  withdraw  annually? 

(c)  If  he  desired  to  have  $10,000.00  left  at  the  end 
of  the  time,  what  would  be  his  annual  allowance? 

182.  300  members  of  the  graduating  class  of  a  college 
plan  to  present  their  college  with  a  scholarship  fund 
of  principal  $100,000.00  on  the  twentieth  anniversary 
of  their  graduation.  They  propose  to  accumulate 
this  fund  by  annual  payments,  the  first  payment 
to  be  made  one  year  after  graduation.  If  the  fund 
can  be  accumulated  at  4%  annually,  what  is  the 
annual  payment  necessary  from  each  member? 

183.  What  would  be  the  annual  payment  in  the  above 
problem  if  the  first  payment  were  made  on  the  date 
of  graduation,  and  the  last  on  the  twentieth  anni- 
versary? 

184.  Explain  the  meaning  of  the  terms  cable,  demand, 
and  60  day,  used  in  quoting  sterling  exchange.  If 
the  cable  rate  is  4.20,  and  interest  is  reckoned  at  4%, 
exact  time,  state  the  demand  and  60  day  rates. 
Allow  15  days  for  transit. 

185.  A  debt  of  $200,000.00  is  to  be  paid  at  the  end  of 
twenty-five  years  by  means  of  a  sinking  fund  into 
which  annual  payments  are  made.    If  the  first  pay- 


REVIEW  PROBLEMS  135 

ment  is  made  one  year  from  the  date  when  the  debt 
is  incurred,  and  all  sums  are  invested  at  an  average 
rate  of  43^%  annually,  what  is  the  amount  of  the 
annual  payment? 

186.  How  much  can  a  man  afford  to  spend  on  a  piece  of 
equipment  which  must  be  replaced  every  five  years 
at  a  cost  of  $1,000.00  in  order  to  get  better  equipment 
which  will  last  eight  years?    Assume  4%  annually. 

187.  A  plant  consists  of: 

X  cost  $20,000.00  scrap  value  $2,000.00  life  18years 
Y  11,000.00  1,500.00         15 

Z  30,000.00  none  5 

Find  the  annual  depreciation  charge  on  a  4%  annual 
basis,  sinking  fund  method. 

188.  Find  the  composite  life  of  the  above  plant. 

189.  A  bank  loans  a  farmer  $3,000.00  to  be  repaid  with 
interest  at  5  3/^%  nominal,  payments  and  interest 
semiannual  for  fifteen  years.  Find  the  semiannual 
payment. 

190.  A  man  was  killed  in  an  accident,  and  an  industrial 
commission  awarded  his  wife  compensation  to  the 
amount  of  $5,000.00,  but  suggested  that  the  amoimt 
be  paid  at  the  rate  of  $50.00  a  month.  On  a  3% 
effective  basis,  how  long  will  the  payments  con- 
tinue? j 

191.  A  man  begins  at  the  age  of  twenty-five  to  save  $10.00 
a  month  and  invest  his  savings  in  a  bank  which  pays 
5%  nominal,  compounding  quarterly.  How  much 
will  he  have  when  he  is  sixty  years  old? 

192.  If  the  beneficiary  of  a  life  insurance  policy  for 
$5,000.00  chooses  to  accept  settlement  in  twenty-five 
annual  payments,  the  first  to  be  made  at  once,  what 
is  the  annual  payment  at  3%  annually? 


136  MATHEMATICS  FOR  THE  ACCOUNTANT 

193.  A  5%  JJ  bond  redeemable  at  par  $1,000.00  twenty- 
eight  years  from  date  is  advertised  for  sale  at  92 
"to  yield  about  5.60".  How  nearly  correct  is  this 
statement?    Is  the  yield  above  or  below  5.60? 

194.  A  banker  wished  to  remit  to  his  correspondent  in 
Florence,  Italy,  55,000  lire.  Direct  exchange  was 
at  7.05.  He  bought  $7,500  in  pesetas,  exchange  at 
.203,  and  instructed  his  Madrid  correspondent  to 
change  them  to  lire  after  deducting  3^%  commis- 
sion. Madrid  quoted  lire  at  67.50.  (That  is,  100 
lire  for  67.50  pesetas). 

a)  Was  the  remittance   sufficient;    what  was  the 
margin  or  deficit  in  lire? 

b)  If  the  remittance  was  insufficient,  what  should 
the  banker  have  paid  in  dollars? 

c)  Which  was  more  profitable:  direct  exchange  or 
arbitration? 

195.  What  sum  paid  at  the  end  of  each  year  for  five  years 
will  extinguish  three  debts,  $1,000.00  due  in  three 
years,  $650.00  due  in  one  year,  and  $475.00  due  in 
four  years.    Base  on  5%  annually. 

196.  Which  is  more  profitable:  to  rent  an  ocean-going 
steamer  for  $30,000.00  per  year  for  twenty  years, 
payable  in  advance,  or  to  buy  it  for  $400,000.00, 
assuming  that  it  will  be  worth  $5,000.00  as  scrap  at 

the  end  of  the  twenty  years?    Base  on  4%  annually. 

197.  A  debt  of  $8475.00  bearing  interest  at  4%  annually 
is  being  repaid  in  annual  instalments  of  $1,000.00. 
In  how  many  years  will  the  debt  be  discharged? 

198.  A  bond  of  par  $1,000.00,  5%  JJ,  redeemable  at  105 
on  January  1,  1921,  is  bought  on  January  1,  1917,  to 
yield  8%.  Find  the  price  and  set  up  a  schedule  of 
accumulation  of  discount. 


REVIEW  PROBLEMS  137 

199.  How  much  must  be  given  to  endow  a  motion 
picture  outfit  costing  $3,000.00,  scrapValue  $500.00 
at  the  end  of  eight  years,  annual  operating  expense 
$2,000.00,  on  a  5%  annual  basis? 

200.  A  5%  JJ  bond  redeemable  at  par  $1,000.00  on 
January  1,  1918,  is  purchased  May  1,  1911,  to  yield 
4%.    What  is  the  price? 

201.  A  woman  who  has  funds  on  deposit  in  a  savings  bank 
which  pays  4%  and  compounds  quarterly  is  con- 
sidering the  purchase  of  bonds  at  par,  $10,000.00, 
which  will  net  4.15%  annually.  Assuming  that  the 
bond  interest  will  be  paid  promptly  and  that  she  will 
deposit  it  at  once  in  the  bank  mentioned,  what  is 
the  difference  on  the  return  for  the  first  ten  years 
on  a  deposit  of  $10,000.00? 

202.  Change  f  25,000  to  United  States  money,  exchange 
at  5.165^. 

203.  If  a  mechanical  piano  for  a  dance  hall  costs  $2,200.00, 
and  saves  $100.00  a  month  in  wages  of  a  pianist 
and  repairs  to  the  piano,  in  what  time  will  the 
mechanical  piano  pay  for  itself  on  a  6%  effective 

basis. 

204.  A  farm  yields  an  average  annual  crop  of  $5,000.00. 
The  annual  expenses  are  $485.00  for  equipment, 
$350.00  for  fertihzer,  and  $2,775.00  for  wages, 
board  and  so  on.  Find  a  fair  cash  price  for  the 
farm  at  6%  annually. 

205.  Change  $1,250.00  to  francs,  exchange  at  6.57^. 

206.  Set  up  a  schedule  showing  the  reserve  against  a 
truck  costing  $3,500.00,  scrap  value  $500.00  at  the 
end  of  five  years,  by  the  method  of  fixed  percentage 
on  a  diminishing  book  value. 


138  MATHEMATICS  FOR  THE  ACCOUNTANT 

207.  A  manufacturing  plant  has  the  following  assets: 

A  wearing  value  $40,000.00  life  50  years 
B  25,000.00        25 

C  5,750.00        18 

D  6,750.00        20 

E  2,400.00  5 

What  is  the  annual  depreciation  charge  on  a  4%  annual 
basis,  sinking  fund  method? 

208.  Change  £382  14s.  7d.  to  dollars,  exchange  at 
4.83875. 

209.  A  corpoiation  issues  bonds  for  $200,000.00,  par 
$1,000.00,  paying  5%  J  J,  redeemable  at  par  in  ten 
years.  Is  it  better  for  the  company  to  pay  the 
semiannual  interest  and  accumulate  a  sinking  fund 
by  semiannual  payments  improved  at  4%  nominal, 
compounding  semiannually,  to  retire  the  bonds  at 
maturity;  or  to  buy  bonds  in  the  open  market  on 
each  interest  date  at  an  average  price  of  105  through- 
out the  ten  years,  so  that  the  semiannual  charges 
for  redemptions  and  interest  on  outstandings  shall 
be  as  nearly  equal  as  possible. 

210.  $1,000.00  is  invested  in  a  60  day  bill  on  Paris  ex- 
change at  9.50.  At  maturity  the  bill  is  sold  at  9.20. 
If  money  is  worth  6%,  banker's  time,  what  is  the 
net  profit  on  the  investment. 

211.  A  debt  of  $10,000.00  is  to  be  amortized  by  ten  equal 
biennial  payments.  If  money  is  worth  6%  annual, 
what  is  the  biennial  payment? 

212.  A  charitable  organization  receives  a  bequest  of 
$5,000.00  with  the  understanding  that  it  is  all  to 
be  used  within  three  years.  On  a  ^%%  annual 
basis,  what  is  the  annual  income? 

213.  Find  the  equated  date  for  these  transactions  at 
6%  exact  time: 


REVIEW  PROBLEMS  139 

Debits:    May  11,  $500.00;  May  19,  82,000.00,  60 
days:  June  2,  90-day  note  for  $500.00. 
Credits:  May  24,  $350.00;  June  12,  $850.00. 

214.  A  5%  bond,  AO,  par  $1,000.00,  redeemable  October 
1,  1920,  at  par,  is  purchased  October  1,  1915,  at 
$950.00.    What  is  the  rate  of  income? 

215.  What  is  the  price  of  the  above  bond  to  yield  6%? 

216.  Prepare  a  schedule  for  the  bond  of  problem  215, 
changing  to  a  December  31-June  30  basis,  and 
carrying  to  maturity. 

217.  A  man  pays  an  annual  premium  of  $33.38  on 
an  endowment  insurance  policy.  Payments  are  of 
course  in  advance.  At  the  end  of  twenty  years 
the  insurance  company  pays  him  $1,000.00.  What 
simi  deposited  semiannually  in  a  savings  bank  which 
pays  43^%  nominal,  compounding  semiannually, 
would  have  accumulated  to  the  same  amount? 
Assume  payments  at  the  beginning  of  the  period. 
On  this  basis  how  much  of  each  payment  was  the 
cost  of  the  endowment  as  an  investment,  and  how 
much  was  cost  of  insurance  and  loading? 

218.  "WTiat  is  the  better  yield :  A  4%  bond  redeemable  at 
par,  bought  at  95;  or  a  5%  bond  redeemable  at  110, 
bought  at  102. 

219.  What  is  the  value  in  English  money  of  a  90-day  bill 
for  $1,500.00  when  the  demand  rate  is  4.6875,  and 
the  discount  rate  is  33^^%? 

220.  Which  is  better  for  an  American  importer  of  English 
goods:  a  quotation  of  4.65  on  London;  or  5.173^  on 
Paris  and  25.40  in  Paris  on  London?  The  face  of 
the  invoice  is  £10,000. 


140  MATHEMATICS  FOR  THE  ACCOUNTANT 

TWO  EXAMINATION  PAPERS 
Do  any  six  examples,  in  any  order.:  Tables  may  be  used. 

1.  Find  the  balance  due  on  the  following  account  at 
December  1,  on  a  5%  basis,  exact  time: 

Debits:  July  7,  S500.00,  60  days;  August  8,  S600.00, 
60  days;  October  5,  $500.00,  30  days;  November  12, 
$450.00,  30  days. 

Credits:  August  9,  60-day  note  without  interest, 
$1,000.00;  November  1,  cash,  $400.00;  November  20, 
cash,  $200.00. 

2.  Balance  the  following  account  as  of  October  1,  without 
interest.  State  the  balance  both  in  dollars  and 
sterling,  exchange  at  4.85^.  State  the  apparent 
profit  or  loss  for  the  month. 

Debits:  Sept.  5,  £350  @  4.86^;  Sept.  9,  £225  8s.  5d. 
@  4.86;  Sept.  21,  £310  6s.  @  4.86)^. 
Credits:    Sept.  8,   £300  @  4.86 M;  Sept.  16,   £405 
10s.  5d.  @  4.8615. 

3.  What  is  the  cost  of  the  draft  in  payment  of  an  invoice 
amounting  to  f  15,000,  exchange  at  7.453^. 

4.  Find  the  amount  and  present  worth  of  a  10-year 
annuity  due  of  $200.00  a  year  at  33^%  annually. 

5.  A  concern  issues  $200,000.00  of  serial  bonds  bearing 
interest  at  6%  annually.  The  trust  agreement  pro- 
vides that  an  annual  payment  of  $20,000.00  shall  be 
made  to  meet  interest  charges  on  outstandings  and 
to  retire  as  many  bonds  as  possible.  In  what  time 
will  the  entire  series  be  retired.?  Solve  without 
schedule. 

6.  A  coi-poration  plans  to  accumulate  a  sinking  fund  to 
retire  a  bond  issue  of  $250,000.00  maturing  at  the  end 
of  25  years.  If  the  trustee  can  keep  the  fund  invested 
at  an  average  rate  of  4%  nominal,  compounding  semi- 


REVIEW  PROBLEMS  141 

annually,  what  amount  must  the  corporation   pay 
him  at  the  end  of  each  six  months? 

7.  In  the  preceding  problem  what  amount  ought  the 
trustee's  books  to  show  at  the  end  of  the  fifth  year? 
Solve  without  schedule. 

8.  A  machine  costing  $2,000.00  has  an  estimated  life  of 
12  years  and  scrap  value  $500.00.  The  reserve  is  to 
be  built  up  by  the  method  of  fixed  percentage  on 
diminishing  book  value.     Show  the  percentage. 

9.  A  20-year  bond  redeemable  at  par  $1,000.00  on 
January  1,  1939,  bearing  interest  at  6%  J  J,  is  pur- 
chased January  1,  1919,  to  yield  5%.  What  is  the 
price? 

10.  A  $1,000.00  bond  bearing  semiannual  coupons  at  the 
rate  of  53^%,  is  redeemable  at  110  after  18  years. 
Find  the  price  to  yield  4%. 

1.  On  July  1  your  account  with  your  Chicago  corres- 
pondent shows: 

Debits:"  March  8,  $500.00,  60  days;  May  6,  $500.00, 

60  days;  June  5,  $250.00,  10  days. 

Credits:  April  8,  $500.00,  60  days;  June  24,  $300.00, 

cash. 

Settlements  are  made  semiannually  at  6%  exact  time. 

State  the  balance  as  of  July  1. 

2.  On  what  date  could  the  above  account  be  settled  with 
a  minimum  payment  of  interest;  that  is,  what  is  the 
equated  date? 

3.  a)    Change  $1,200.00  to  sterling  at  4.6565. 

b)    At  the  same  quotation  change  £255  13s.  7d.  to 
dollars. 

4.  You  are  importing  olive  oil,  and  have  these  quotations 
per  hundred  litres: 


142  MATHEMATICS  FOR  THE  ACCOUNTANT 

From  Florence,  600  lire,  exchange  at  7.55. 
From  Marseilles,  425  francs,  exchange  at  5.70. 
Which  is  cheaper,  and  how  much? 

5.  Find  the  amount  of  a  15-year  annuity  of  $1.00  a 
year  accumulated  at  4^%  annually. 

6.  A  machine  costing  $3,000.00  has  a  life  of  four  years 
and  scrap  value  $200.00. 

a)  Find  the  annual  depreciation  charge  by  the  sink- 
ing fund  method,  at  4%  annually .- 

b)  Find   the  annual   rate   of  depreciation  by   the 
method  of  fixed  rate  on  diminishing  book  value. 

7.  I  have  $10,000.00  in  a  Trust  Company  which  pays 
4%  nominal,  compounding  semiannually.  Shall  I 
draw  the  money  and  buy  a  6%  stock  at  125,  dividends 
payable  semiannually,  and  invest  the  dividends  in 
the  same  Trust  Company.  Which  is  the  better  in- 
vestment and  how  much  over  a  ten-year  period. 

8.  Value  a  $1,000.00  bond,  redeemable  after  13  years 
at  par,  on  a  7%  basis.    Interest  on  the  bond  6%  J  J. 

9.  Value  a  4%  AO  bond,  par  $1,000.00,  redeemable  at 
108  after  8  years,  to  yield  6%. 

10.  A  series  of  $1,000,000.00  of  bonds,  par  $1,000.00, 
bearing  interest  at  5%  annually,  is  redeemable  at  par 
$50,000.00  each  year  from  January  1,  1921,  to  1940. 
Value  this  series  as  of  January  1,  1920,  on  a  4% 
annual  basis. 


CHAPTER  XIV 

PROBLEMS    FROM    EXAMINATIONS    OF    THE 

AMERICAN  INSTITUTE  OF  ACCOUNTANTS 

This  chapter  is  intended  to  summarize  briefly  such 
parts  of  the  book  as  are  most  useful  to  accountants  who 
are  preparing  for  the  examinations  of  the  American 
Institute  of  Accountants.  The  student  should  first  read 
the  paragraphs  on  reversed  multiplication  and  division 
in  chapter  I.  These  shortened  processes  are  valuable 
time  savers  in  the  rush  of  an  examination,  and  the 
results  obtained  by  them  are  sufficiently  accurate.  The 
chapter  on  averaging  accounts  should  be  read,  as  there 
was  such  a  problem  on  a  recent  examination,  and  even 
the  matter  of  equated  date  is  sometimes  required  on 
various  C.  P.  A.  papers. 

The  first  matter  of  importance  which  has  not  been 
taken  up  in  the  body  of  the  book  is  the  averaging  of  an 
account  with  an  English  correspondent.  This  problem 
occurred  on  the  Institute  examination  in  November,  1918. 

A  dealer  in  foreign  exchange  finds  from  his  books  that 
he  has  had  the  following  transactions  in  London  exchange 
during  a  particular  month,  viz. : 
Exchange  bought  in  the  local  market : 

Jan.    1,  30-day  bill  payable  in  London  £300  @  4.75. 
15,  bill  due  at  sight  in  London,  £2,500  @  4.76. 
Exchange  sold  in  the  local  market: 

Jan.    5,  bill  due  in  London  at  sight  £1,000  @  4.77. 
20,  cable  transfer,  £2,000  @  4.78. 
Foreign  correspondent's  draft  honored  and  paid : 

Jan.  20,  bill  at  30  days  after  sight  accepted  Dec.  21, 
£500  @  4.78. 

143 


144  MATHEMATICS  FOR  THE  ACCOUNTANT 

State  how  the  balance  stands  on  the  account  at  the  close 
of  the  month,  and  how  much  profit  or  loss  has  been 
derived  from  the  transactions.  (At  Jan.  31  the  rate  for 
cable  transfers  was  4.78).  Is  the  profit  or  loss  so  stated 
final? 

The  solution  in  the  Journal  of  Accountancy  for  Feb- 
ruary, 1919,  gave  the  following  form  as  the  usual  one  in 
the  United  States.  The  transactions  are  listed  in  both 
currencies,  the  sterling  being  used  as  an  inventory.  This 
inventory  is  valued  at  the  end  of  the  month,  and  the  profit 
or  loss  determined  from  this. 

London  Correspondent 

Debit      Jan.    1,  £    300  @  4.75      $  1,425 

15,     2,500  @  4.76        11,900 

20,        500  @  4.78  2,390 

31,  Cr.  Exchange  a/c  4i) 

£3,300  S15,764~ 


Credit     Jan.    5,  £1,000  @  4.77        $4,770 

20,     2,000  @  4.78         9,560 

31,        300  @  4.78  bal.  1,434 

£3,300  S15,764 

The  balance  of  sterling  on  hand  January  31  was  £300; 
the  value  of  this  at  4.78  was  SI, 434.  This  item  is  entered 
on  the  credit  side.  Then  the  dollar  columns  are  footed 
and  balanced  as  usual,  and  there  is  found  to  be  a  profit 
of  S49.00,  which  is  transferred  to  Exchange  account. 

The  following  comment  is  made  in  answer  to  the  last 
question:  "The  profit  is  not  final.  The  balance  may  not 
realize  4.78.  Besides  there  is  the  interest  on  the  thirty- 
day  bill,  and  on  the  overdraft  caused  by  the  credit  of 
January  20  which  will  not  be  covered  until  the  sight  draft 
of  January  15  reaches  London.    Also  the  London  corres- 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.     145 

pondent  may   charge   a   commission   for  handling   the 
business." 

A  similar  problem  occurred  on  the  Massachusetts 
C.  P.  A.  examination  for  1914.  The  problem  will  be 
given  without  solution. 

A  banking  concern  dealing  in  foreign  exchange  has  the 
following  transactions  on  its  account  with  its  London 
correspondent: 

Debits : 
Sept.     1,  remittance,  30-day  bill  £400  @  4.86. 
10,  remittance,  sight  bill  £100  10s.  @  4.87. 
15,  remittance,  demand  bill  £200  Os.  6d.  @  4.8G75 

Credits : 
Sept.    2,  sight  draft  £300  @  4.873^. 

12,  demand  draft,  £200  12s.  5d.  @  4.87. 
20,  cable,  demand,  £100  @  4.88. 
Ascertain  the  profit  or  loss  on  the  account  for  the  month 
of  September  and  state  the  balance  as  of  Sept.  30,  in 
foreign  and  domestic  currency,  the  current  rate  on  that 
date  being  4.89. 

Compound  interest:  The  simplest  method  of  arriving 
at  the  compound  interest  on  a  given  amount  for  a  given 
time  and  rate  is  to  compute  the  compound  amount  first 
and  deduct  the  principal.  All  such  problems  are  worked 
on  a  basis  of  one  unit:  one  dollar,  one  pound  sterling, 
one  franc,  and  so  on.  One  unit  at  5%  for  one  unit  of 
time,  whether  a  year,  six  months  or  other  period,  earns 
.05  of  itself,  and  so  at  the  end  of  the  period  amounts  to 
1.05  of  itself.  This  new  principal  during  the  next  unit 
of  time  earns  .05  of  itself,  and  at  the  end  of  the  period 
amounts  to  1.05  of  its  value  at  the  beginning  of  the  period. 
The  simplest  way  to  arrive  at  this  amount  for  two  periods 
is  therefore  to  multiply  1.05,  the  amount  at  the  end  of 
the  first  period,  by  1.05,  the  ratio  of  increase  during  the 


146  MATHEMATICS  FOR  THE  ACCOUNTANT 

second  period.  Further,  note  that  the  ratio  of  increase 
during  any  period  is  1.05;  that  is,  the  amount  at  the  end 
of  any  period  is  1.05  of  the  amount  at  the  beginning.  To 
find  the  amount  at  the  end  of  the  sixth  period,  for  in- 
stance, multiply  1.05  together  6  times.  Algebraically 
expressed,  this  is  (1.05)  ^  which  we  read  1.05  to  the  sixth 
power.  In  multiplying,  it  is  of  course  wise  to  use  reversed 
multiplication.  The  1.05  is  called  the  base,  and  6  is  called 
the  exponent,  showing  how  many  times  the  base  must  be 
used  as  a  fact    . 

Another  shortcut:  It  is  not  necessary  to  multiply  five 
times  to  obtain  this  result.  When  we  have  multiplied 
1.05  X  1.05  X  1.05  we  have  obtained  (1.05)3.  Now  if  we 
multiply  this  result  by  itself,  we  shall  be  in  effect  multi- 
plying by  1.05  X  1.05  X  1.05  all  over  again,  except  that 
we  multiply  once  instead  of  three  times.  This  leads  to  an 
important  algebraic  truth  which  was  necessary  in  answering 
a  question  which  occurred  on  the  1917  examination.  The 
truth  referred  to  is  this:  since  (1.05)3X(1.05)3=(1.05)«,  m 
general  whenever  exponents  are  added,  bases  are  multiplied. 

The  problem  was  this:  You  are  called  upon  to  state 
what  is  the  annual  sinking  fund  payment  necessary  to 
redeem  a  principal  of  $1,000,000.00  due  30  years  hence, 
it  being  assumed  that  the  annual  sums  set  aside  are  in- 
vested at  compound  interest  at  5%  annually.  State 
what  computation  you  would  make  to  arrive  at  the 
result  desired.    You  need  not  work  out  the  computation. 

The  solution  of  this  problem  requires  the  computation 
of  (1.05)^".    The  shortest  solution  is: 
Find  (1.05)2 
That  quantity  multiplied  by  itself  gives  (1.05) 

rt  u  u  u  u  «         (1,05)8 

U  u  u  a  u  u        (1,05)^^ 

a  a  u  u  u  (I        (1.05)^2 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.    147 

This  result  is  two  periods  more  than  the  required  time. 
Since  adding  exponents  is  the  same  as  multiplying  bases, 
evidently,  subtracting  exponents  is  the  same  as  dividing 
bases.    So  (1.05)  =«  2  divided  by  (1.05)  ^  will  give  (1.05) »°. 

The  finding  of  the  sinking  fund  payment  will  be  shown 
later. 

To  return  to  the  matter  of  compound  amount,  the 
simplest  method  of  working  out  the  amount  of  one  unit 
for  any  rate  and  time  is  to  multiply  1  plus  the  rate  of 
interest  by  itself,  or  to  multiply  multiples  of  this  quantity 
together,  until  the  desired  total  of  exponents  is  obtained. 
This  compound  amount  is  often  indicated  by  the  symbol 
s  °.  This  quantity  is  usually  given  in  problems  where  it 
must  be  used  in  finding  a  sinking  fund  payment,  or 
summing  an  annuity,  or  in  other  cases. 

Compound  interest  for  any  period  is,  as  has  been  said, 
amount  less  principal.  In  many  problems  the  interest  is 
in  question  rather  than  amount,  so  this  fact  should  be 
kept  in  mind. 

Present  worth:  One  very  common  use  for  compound 
amount  is  in  finding  present  worth.  If  a  dollar  will 
amount  in  five  years  at  5%  to  $1.27628156,  then  the 
principal  required  to  accumulate  $1.00  at  the  end  of  five 
years  at  5%  must  be  1  divided  by  1,27628156,  or 
$.78352617.  In  general,  if  1  amounts  in  n  years  at  rate 
i  to  (1+i)  "^j  ors  °,  then  the  present  investment  which  will 

yield  1  is  ^ —  or  —   The  rule  for  finding  present  worth 

(1+1)°,      s°. 

then  is  to  divide  compound  amount  into  1.  The  symbol 
for  this  quantity  is  v  °.  It  is  usually  given  when  it  is 
necessary  in  working  out  an  amortization  factor.  But 
if  s  °  happens  to  be  given,  v  °  can  easily  be  found  by 
division.  Contracted  division  should  be  used  in  this 
computation. 


148  MATHEMATICS  FOR  THE  ACCOUNTANT 

Compound  discount  is  simply  the  discount  found  by 
compound  interest  methods.  We  are  familiar  with  dis- 
count on  notes,  found  by  simple  interest.  It  is  the 
difference  between  present  value  and  the  par  value  at 
some  future  date.  So  compound  discount  is  the  difference 
between  1  and  the  present  worth  of  1.  Expressed  as  a 
rule,  subtract  the  present  worth  from  the  par  value  due 
at  some  time  in  the  future.  This  quantity  is  important 
in  annuity  calculations.  The  compound  discount  on  1 
for.5  years  at  5%  is  1— .78352617,  or  .21647383. 

Annuities:  An  annuity  is  a  series  of  equal  payments 
made  at  equal  intervals  of  time,  improved  at  a  fixed  rate 
of  interest.  The  rule  for  finding  the  amount  of  an  annuity 
is: 

Divide  the  compound  interest  for  the  given  rate  and 

time  by  the  interest  for  one  period,  and  multiply  by  the 

periodic  payment. 
This  is  demonstrated  with  schedules  at  the  beginning  of 
chapter  VII.  For  instance,  the  amount  of  payments  of  1 
each  at  the  end  of  each  year  for  15  years  accumulated  at 
4%  annually,  given  that  the  compound  amount  of  1  for 
the  same  time  and  rate  is  1.80094351,  is  found  as  follows: 
compound  amount  1.80094351 

compound  interest  .80094351 

single  interest  .04 

quotient  20.02358775 

Note  that  the  payments  are  held  to  be  made  at  the  end 
of  the  period.  This  is  because  in  the  business  world  such 
payments  are  usually  made  out  of  profits,  which  are  reck- 
oned at  the  end  of  the  fiscal  period.  If  the  payments  are 
made  at  the  beginning  of  the  period,  the  result  is  an 
annuity  due,  as  it  is  called.  The  amount  of  such  an 
annuity  can  be  found  from  the  amount  of  an  ordinary 
annuity  by  multiplying  by  1  -f-  i.    In  the  above  annuity. 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.     149 

if  the  payments  were  made  at  the  beginning  of  each  period, 
the  amount  would  be  20.02358775  X  1.04  =  20.82453126. 

The  present  worth,  or  cash  value,  of  an  annuity  is  a 
term  used  in  two  senses.  It  may  mean  first  the  cash  pay- 
ment which  will  extinguish  a  debt  which  was  intended  to 
be  repaid  in  annuity  form  with  inteiest  charged  on 
balances  outstanding.  Second  it  may  mean  the  present 
investment  which  will  yield  a  given  income;  the  interest 
on  the  portion  remaining  invested  increases  that  protion 
and  therefore  increases  the  term  during  which  the  income 
can  continue.  This  last  is  the  annuity  in  the  banking  and 
insurance  sense.  The  rule  for  valuing  the  present  worth 
of  an  ordinary  annuity  is : 

Divide  the  compound  discount  for  the  given  time  and 

rate  by  the  interest  for  one  period,  and  multiply  by  the 

periodic  payment. 
This  is  also  demonstrated  with  schedules  in  chapter  VII. 
For  instance,  the  present  worth  of  the  annuity  which  we 
summed  just  above,  is  found  as  follows,  given  that  the 
present  worth  of  1  for  15  years  at  4%  annually  is  .5552645: 
present  worth  .5552645 

compound  discount  .4447355 

single  interest  .04 

quotient  11.1183875 

The  present  worth  of  an  annuity  due  is  found  by  multi- 
plying the  present  worth  of  an  ordinary  annuity  by  1  +  i. 
If  the  above  were  an  annuity  due  its  present  worth  would 
by  11.1183875  X  1.04  =11.563123. 

One  more  sort  of  annuity  which  occurs  frequently  is 
the  deferred  annuity.  This  is  an  aiuiuity  of  n  payments 
which  does  not  begin  until  after  m  periods;  that  is,  the 
first  payment  is  to  be  made  at  the  end  of  the  first  period 
after  m  periods  have  expired.  For  instance,  the  above 
15-year  annuity  might  be  deferred  5  years.    This  means 


150  MATHEMATICS  FOR  THE  ACCOUNTANT 

that  the  first  payment  is  made  at  the  end  of  the  6th  year. 
There  are  two  methods  of  finding  the  present  worth  of 
such  an  annuity. 

We  have  already  found  that  at  the  beginning  of  the 
sixth  year,  which  is  the  same  as  the  end  of  the  fifth  year, 
the  present  worth  is  11.1183875.  Since  this  amount  is 
not  due  for  five  years,  it  must  be  discounted  to  the  present 
time.  The  proper  method  of  doing  this  is  to  multiply 
by  the  present  worth  of  1  for  5  years  at  4%.  If  the 
problem  does  not  state  this  we  must  proceed  as  in  the 
model  problem.  The  present  worth  is  1  divided  by  the 
compound  amount.  By  multiplication  we  find  this  com- 
pound amount  to  be  1.2166529,  which,  divided  into  1, 
gives  as  quotient  the  present  worth,  .82192711.  Now 
multiply  this  by  11.1183875,  and  we  have  the  worth  of 
this  15-year  annuity,  which  will  not  begin  for  five  years; 
this  value  is  9.13850408. 

The  other  method  is  to  subtract  the  present  worth  of 
an  annuity  for  the  term  of  deferment  from  the  present 
worth  of  an  annuity  from  the  present  time  to  the  time 
of  the  last  payment.  For  the  annuity  we  are  discussing, 
subtract  the  present  worth  of  a  5-year  annuity  from  that 
of  a  20-year  annuity. 

If  present  worth  of  a  20-year  annuity  is  13.59032634 
and  of  a  5-year  annuity  4.45182233 

subtracting,  we  have  9.13850401 

the  present  worth  of  a  15-year  annuity  deferred  5  years, 
interest  at_4%  annually. 

The  following  Institute  problem  included  both  amount 
and  present  worth  of  an  ordinary  annuity: 
A  owns  an  annuity  of  $50.00  per  annum,  the  first  payment 
on  which  falls  due  one  year  hence,  and  continues  for  a 
period  of  20  years  certain.    State 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.     151 

a)  the  present  value  of  the  benefit; 

b)  the  amount  which  he  will  have  accumulated  at  the 
end  of  the  period  if  he  invests  each  moiety  as  it 
becomes  due. 

Assume  interest  at  4%  payable  annually.  In  this  con- 
nection it  is  stated  that  the  value  of  1.04  =">  is  2.191123. 

,  ,  ,,  compound  discount 

a)  present  worth    =    

single  interest 

compound  amount  2191123 

by  division  .456387  is  present  worth 

compound  discount  .543613 

divided  by  .04 

present  worth  of  an 

annuity  of  1  13.5904 

times  50  $679.52 

If  only  the  present  worth  had  been  required  very  likely 
the  present  worth  of  1,  namely  v^o,  would  have  been 
given.  In  that  case  the  first  division  would  have  been 
avoided,  and  the  solution  would  begin  with  finding  the 
compound  discount. 

,  ,  ,       compound  interest 

b)  amount  = -. — - — : 

smgle  mterest 

compound  amount  2.191123 

compound  interest  1.191123 

single  interest  .04 

quotient  29.778075 

times  50  $1,488.90 

On  the  May,  1919,  examination  occurj-ed  this  problem: 
A  lease  has  run  five  years  to  run  at  $1,000.00  a  year  pay- 
able at  the  end  of  each  year,  with  an  extension  for  a 
further  five  3 ears  at  $1,200.00  a  year.  On  a  6%  basis 
what  sum  should  be  paid  now  in  lieu  of  the  ten  year's 
rent?     v^  at  6%  =  .7473. 


152  MATHEMATICS  FOR  THE  ACCOUNTANT 

for  the  first  five  years: 
present  worth  .7473 

compound  discount  .2527 

divided  by  .06  4.211667 

times  1,000  $4,211.67 

For  the  last  five  years,  the 
value  at  the  beginning  of 
the  sixth  year  is,  as  above  4.211667 
discount  this  5  years,  mul- 
tiplying by  v^  .7473 
product                                   3.1474 
times  1,200  3,776.85 
total                                                             $7,988.52 
Comment  on  the  solution:    The  present  worth  of  the 
rentals  for  the  first  five  years  is  found  by  the  rule  for 
present  worth:  divide   the  compound  discount  by   the 
single  interest.    The  last  five  years'  rentals  constitute  a 
five-year  annuity  deferred  five  years.    The  present  worth 
at  the  beginning  of  the  sixth  year  is  the  same  per  dollar 
as  for  the  first  five  years.    But  this  must  be  discounted 
from  the  beginning  of  the  sixth  year  to  the  beginning  of 
the  first  year,  multiplying  by  v^    If  more  digits  had  been 
given  in  the  value  of  v^  the  problem  might  have  been 
solved  by  the  second  method.    v^°  =  v^  times  v^;  .7473 
X  .7473  =  .5584,  about. 

present  worth  .5584 

compound  discount  .4416 

divided  by  .06  7.36 

the  present  worth  of  a  10-year  annuity, 
subtract  present  worth  of  a  5-year 

annuity  4.2117 

giving  3.1483 

present  worth  of  the  deferred  annuity, 
times  1,200  =  $3,777.96,  which  is  very  in- 
accurate, owing  to  the  lack  of  sufficient  digits- 


PROBLEMS  FROM  EXAMINATIONS  OF  TliE  A.  I.  A.     153 

Sinking  funds:  We  saw  that  if  1  per  annum  for  15 
years  was  improved  at  4%  annually,  the  sum  at  the  end 
of  the  15  years  was  20.02358775.  Per  contra,  if  it  is 
desired  to  make  an  annual  payment  for  15  years  such 
that  at  the  end  of  the  15  years  the  sum  shall  be  1,  this 
annual  payment  is  1  divided  by  20.02358775,  which  is 
.049941 1.  Now  this  20.02358775  was  obtained  by  dividing 
the  compound  interest  by  the  single  interest.  Evidently 
the  annual  payment  which  will  amount  to  1  should  be 
obtained  by  dividing  the  single  interest  by  the  compound 
interest.  This  annual  payment  which  is  to  amount  to  a 
given  sum  is  used  most  commonly  in  finding  payments 
into  the  various  reserves,  whether  for  sinking  funds  or  for 
depreciation. 

This  problem  occurred  on  a  recent  examination:  In 
auditing  the  books  of  a  corporation  you  find  that  in 
order  to  provide  a  sum  to  redeem  a  mortgage  of 
$100,000.00  faUing  due  at  the  end  of  ten  years,  a  reserve 
of  $8,000.00  per  annum  has  been  set  aside  for  three 
years,  but  that  contrary  to  intention  the  company  has 
failed  to  accumulate  interest  thereon.  Assuming  interest 
at  4%  convertible  annually,  what  should  have  been  the 
total  accumulations  to  date  and  what  amount  should  now 
be  set  aside  for  the  next  seven  years  in  order  to  complete 
the  sinking  fund.    1.04'  =  1.31593. 

The  answer  to  the  first  question  requires  the  summing 
of  a  three-year  annuity  of  $8,000.00  at  4%.  First  we 
must  find  1.04  3;  then 

compound  amount  1.124864 

compound  interest  .124864 

single  interest  .04 

quotient  3.1216 

times  8,000  $24,972.80 


154  MATHEMATICS  FOR  THE  ACCOUNTANT 

This  is  the  amount  which  ought  to  be  in  the  fund.  Whether 
it  is  advisable  to  assume  that  the  interest  of  $972.80  is 
added  to  the  fund  by  an  adjusting  entry  is  questionable. 
There  is  another  question  which  depends  on  the  date 
at  which  the  audit  actually  takes  place.  If  it  took  place 
well  along  in  the  next  fiscal  year,  we  might  assume  that 
the  amount  in  the  fund  would  be  increased  by  one  year's 
interest  and  brought  on  the  books  at  the  end  of  the  year. 
The  remainder  of  the  solution  given  here  assumes  that 
no  interest  was  added,  but  that  the  fund  was  left  at 
$24,000.00,  and  that  the  remaining  payments  must  supply 
the  balance  of  $76,000.00.  The  rule  for  finding  a  sinking 
fund  payment  has  been  given:  divide  the  single  interest 
by  the  compound  interest: 

compound  amount  1.31593 

compound  interest  .31593 

dividing  into  .04 

quotient  .1266 

This  is  the  sinking  fund  payment  per  dollar.    Multiplying 
by  76,000  we  have  the  annual  payment,  $9,621.60. 

On  the  Michigan  C.  P.  A.  examination  in  1915  was  this 
problem : 

A  contractor  proposes  to  build  a  bridge  to  Belle  Isle 
and  accepts  the  city's  4%  20-year  bonds  in  payment  to 
the  amount  of  $2,000,000.00.  He  advocates  as  a  means 
of  retiring  the  bonds  the  establishment  of  a  toll  system 
on  foot  passengers  and  automobiles  at  the  respective  rates 
of  one  and  five  cents  each.  Assuming  the  ratio  of  foot 
passengers  to  automobiles  to  be  ten  to  one,  how  many  of 
each  w^ould  be  necessary  to  pay  the  interest  annually  and 
create  a  fund  which,  placed  at  the  same  rate  of  interest, 
would  be  sufficient  to  retire  the  bonds  at  maturity. 
$1.00  compounded  at  4%  for  20  years  will  amount  to 
2.19112314. 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.     155 

The  solution  will  not  be  given,  but  the  number  of  foot 
passengers  is  9,810,893. 

A  different  type  of  sinking  fund  problem  is  the  follow- 
ing: Argument  has  been  strongly  urged  that,  aside  from 
any  question  of  possible  mismanagement  or  of  the  difiB- 
culty  of  making  satisfactory  investments  to  yield  the 
same  rate  of  interest  as  the  bonds,  a  sinking  fund  for 
bonds  is  more  expensive  than  an  arrangement  for  serial 
repayment  of  the  bonds.  This  is  illustrated  by  the  case 
of  S20,000.00  of  5%  bonds.  If  these  are  paid  off  in  a 
series,  one  each  year,  the  total  payment  will  be  principal 
$20,000.00,  interest  $10,500.00,  total  $30,500.00. 

The  annual  sinking  fund  to  pay  off  these  bonds  would 
on  a  5%  annual  basis  amount  to  $604.85,  making  in  20 
years  $12,097.00,  and  the  interest  paid  on  the  bonds  would 
be  $20,000.00,  total  payments  $32,097.00.  The  apparent 
excess  burden  is  accordingly  $1,297.00.  Discuss  the  above 
argument  and  show  clearly  just  what  the  figures  mean  and 
in  what  the  apparent  saving  actually  consists. 

The  solution  given  in  the  ''Journal  of  Accountancy" 
read  as  follows: 

"If  the  bonds  are  serial,  one  bond  being  paid  off  each 
year,  the  average  capital  of  which  the  company  would 
have  the  use  would  be  $10,500.00.  As  the  interest  is  the 
same  amount,  the  company  paid  100%  in  20  years  or 
5%  per  annum. 

"By  the  sinking  fund  method  the  bonds  were  virtually 
paid  off  at  the  rate  ot  $604.85  per  year.  This  means  that 
the  available  capital  in  use  was  diminished  by  that 
amount  at  the  end  of  each  year — that  is,  that  the  com- 
pany had  the  use  of  $20,000.00  the  first  year,  $19,395.15 
the  second  year,  and  so  on.  As  the  last  payment  was 
made  at  the  end  of  the  twentieth  year,  they  had  the  use 
of  $20,000.00  less  10  X  $604.85,  or  $8,507.85.    This  makes 


156  MATHEMATICS  FOR  THE  ACCOUNTANT 

an  average  capital  in  use  of  $14,253.92.  For  the  use  of 
this  average  capital  the  company  paid  interest  $20,000.00, 
against  which  there  is  a  credit  of  $7,903.00,  the  compound 
inctrest  reahzed  from  the  sinking  fund.  This  means  that 
the  net  interest  charge  was  $12,097.00.  $12,097.00  for 
the  use  of  $14,253.92  means  a  rate  of  a  trifle  less  than 
84.87%  for  twenty  years,  or  4.1435%  per  annum.  The 
advantage  of  the  sinking  fund  method  is  apparent,  and 
is  explained  by  the  fact  that  the  fund  earns  compound 
interest. 

"The  apparent  excess  burden  mentioned  in  the  problem 
is  not  a  true  excess.  It  is  reached  by  calling  the  $20,000.00 
paid  for  coupons  all  interest,  which  is  not  true.  Of  this 
amount  $7,903.00  was  applied  to  the  principal,  being 
the  difference  between  the  face  of  the  bonds,  $20,000.00, 
and  the  actual  payments  to  the  sinking  fund,  $12,097.00. 
The  true  amount  of  interest  was  therefore  $12,097.00,  not 
$20,000.00.  The  interest  paid  must  be  considered  in 
relation  to  the  capital  of  which  the  corporation  had  the 
use  during  the  20  years,  not  in  relation  to  the  face  of 
the  bonds." 

This  closes  the  quotation.  The  solution  does  not  re- 
quire any  computation,  but  does  bring  out  the  idea  of 
average  capita),  and  is  an  interesting  comparison  of  re- 
demption oy  smking  fund  and  serial  redemption  in  equal 
amounts.  If,  however,  the  redemption  had  been  on  an 
amortization  principle,  by  which  the  annual  total  pay- 
ments of  interest  and  principal  together  had  been  as 
nearly  equal  as  possible,  there  would  be  practically  no 
difference  in  dollars  and  cents  between  amortization  and 
the  sinking  fund  redemption. 

Depreciation:  As  has  been  said,  the  depreciation  re- 
serve is  usually  built  up  on  the  sinking  fund  principle. 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.     157 

The  determination  of  the  periodic  payment  under  those 
conditions  would  be  by  the  process  just  demonstrated. 

Another  method  of  reckoning  depreciation  is  by  com- 
puting affixed  percentage  periodically,  based  on  a  dimin- 
ishing balance.  After  the  percentage  has  been  determined, 
the  annual  deduction  from  the  net  book  value  as  at  the 
beginning  of  the  year  can  be  easily  found.  This  method 
is  sometimes  required  in  accounting  problems,  but  the 
percentage  is  fixed  arbitrarily,  and  has  no  reference  to 
any  estimated  life  of  the  asset.  There  is  a  formula  by 
which  the  percentage  can  be  computed,  but  the  com- 
putation requires  the  use  of  logarithms  unless  the  figures 
in  the  problem  are  "doctored".  The  following  problem 
illustrates  this: 

A  machine  costing  $81.00  is  estimated  to  have  a  life  of 
four  years  with  a  residual  value  of  $16.00.  Prepare  a 
statement  showing  the  annual  charge  for  depreciation 
under  each  of  these  methods: 

a)  straight  line; 

b)  constant  percentage  of  diminishing  value ; 

c)  annuity  method. 

For  convenience  in  arithmetical  calculation  assume  the 
rate  of  interest  to  be  10%. 

a)  The  wearing  value,  as  it  is  called,  is  $65.00;  that  is, 
that  amount  of  capital  is  sunk  in  the  machine  and  will 
never  be  reahzed.  In  a  Hfe  of  four  years,  the  annual 
consumption  of  capital  by  the  straight-line  method  is 
1/4  of  that  amount,  or  $16.25. 

b)  The  formula  for  determining  the  constant  percent- 

1  /  g 
age  is  r  =  1  —  V  — 
C 

Expressed  as  a  rule:   Divide  the  scrap  value  by  the  cost; 

find  the  number  which,  multiplied  together  n  times,  will 

give  that  amount;  subtract  from  1. 


158  MATHEMATICS  FOR  THE  ACCOUNTANT 

In  this  problem,  S  is  16;  C  is  81;  n  is  4. 
16  _^      2      ^     ^ 
sT  •"  3  ^  3  ^  3  ^  3 

16       2 
so  the  fourth  root  of  —  is  -,  or  as  a  decimal. 

81        3 

.66  V  3,  or  as  a  percent,  662/3%. 

Subtracting  from   1,   we  have  the  annual  rate. 

33V3%. 

It  is  well  to  subjoin  this  schedule: 

Schedule  Showing  Depreciation  Charged  Against 

Asset ,  Costing  S81.00,  Life  4  Years, 

Estimated  Scrap  Value,  $16.00 
Charged  at  33V/ 3%  Annually  Based  on  Net  Book 

Value  at  Beginning  of  Each  Year 


Value  at 

Amount  in 

Beginning  of 

Depreciation 

Reserve  at 

ear 

Year 

33V/ 3% 

End  of  Year 

i 

$81.00 

$27.00 

$27.00 

2 

54.00 

18.00 

45.00 

3 

36.00 

12.00 

57.00 

4 

24.00 

8.00 

65.00 

'emair 

ider       $16.00  Total      $65.00 

The  computation  by  the  sinking  fund  method  is: 

Compound  amount,  4  years,  at  10%  annually  1.4641 
compound  interest  .4641 

divide  into  .10 

quotient  .21547 

times  65  $14.00555 

The  schedule,  omitting  title,  is: 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.     1^9 


Interest 

Total 

Amount 

on  Fund, 

Annual 

Addition 

in 

ear 

10% 

Payment 

to  Fund 

Fund 

1 

$14.00555 

$14.00555 

$14.0055,1 

2 

$1.4006 

14.00555 

15.40615 

29.41170 

3 

2.9412 

14.00555 

16.94675 

46.35845 

4 

4.6358 

14.00555 

18.64135 

64.99980 

$8.9776  $56.02220  164.99980 
The  annuity  method  assumes  interest  on  the  net  book 
value.  It  is  developed  by  the  following  reasoning:  The 
machine  at  first  cost  a  certain  amount.  At  the  end  of 
the  life  of  the  machine  the  amount  of  capital  that  has  been 
tied  up  is  not  merely  the  original  cost,  but  the  compound 
amount  of  that  cost.  Of  this  compound  amount  a  certain 
part  is  returned  in  the  form  of  scrap  allowance.  The 
balance  is  total  loss,  and  should  be  spread  over  the  life 
of  the  machine  in  the  form  of  a  sinking  fund  charge. 
Expressed  as  a  rule:  Multiply  the  cost  by  the  compound 
amount;  subtract  the  scrap  value;  multiply  by  the  sinking 
fund  factor. 

For  this  machine : 


cost 

$  81.00 

compound  amount  factor 

1.4641 

product 

118.5921 

less  scrap 

16.00 

loss  of  capital 

102.5921 

sinking  fund  factor 

.21547 

product 

22.1055 

The  schedule  is: 

Net  Book 

Annual    Net  Reduc- 

Value  at 

Interest    Depreciation  tion  of 

Year 

Beginning 

10% 

Charge    Book  Value 

1 

$81.00 

$  8.10 

$22.1055      $14.0055 

2 

66.9945 

6.6995 

22.1055        15.4060 

3 

51.5885 

5.1588 

22.1055        16.9467 

4 

34.6418 

3.4642 

22.1055        18.6413 

$16.0005      $23.4225     $88.4220      $64.9995 


160  MATHEMATICS  FOR  THE  ACCOUNTANT 

Amortization:  Just  as  the  sinking  fund  factor  is  the 
reciprocal  of  the  amount  of  an  annuity,  so  the  amortiza- 
tion factor  is  the  reciprocal  of  the  present  worth  of  an 
annuity.  We  saw  that  the  cash  payment  which  would 
yield  an  income  of  one  per  annum  for  15  years  if  interest 
was  allowed  on  balances  at  the  rate  of  4%  per  annum 
was  11.1183875.  If  on  the  other  hand  we  invest  1  now 
and  desire  an  annual  income  for  15  years  on  a  4%  annual 
basis,  evidently  that  annual  income  is  1  divided  by 
11.1183875,  which  is  .0899411.  The  rule  for  finding  the 
present  worth  of  an  annuity  was  to  divide  the  compound 
discount  by  the  single  interest;  so  the  rule  for  finding  the 
amortization  factor  is  to  divide  the  single  interest  by  the 
compound  discount.  But  the  finding  of  compound  dis- 
count may  present  difficulties,  when  the  compound 
amount  is  given,  as  it  was  in  the  problem  on  the  Institute 
paper  of  1918;  because  the  present  worth  must  be  found 
first,  with  possibilities  for  mistakes  in  the  work  and 
probability  of  error  in  the  last  digits;  then  the  compound 
discount  must  be  found  by  subtraction.  But  note  that 
the  amortization  factor  is  .0899411,  while  the  sinking 
fund  factor  is  .0499411,  a  difTerence  of  .04,  the  interest 
rate.  This  fact  can  be  proved  by  algebra,  and  is  in- 
vestigated in  a  schedule  in  chapter  X.  The  simplest  way 
to  find  the  amortization  factor,  given  the  compound 
amount,  is  to  find  the  sinking  fund  factor  first  and  add 
the  interest  rate.  That  method  will  be  employed  in  solv- 
ing the  Institute  problem  referred  to: 

A  corporation  wants  to  retire  a  debt  of  $105,000.00 
bearing  5%  interest  payable  annually.  The  tenth  pay- 
ment, including  interest,  is  to  be  $15,000.00.  The  other 
nine  periodical  payments  are  all  to  include  interest  and 
to  be  of  the  same  amount.  Required  the  amount  of  each 
of  the  nine  payments.    1.05*  =  1.551328. 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.     161 

The  payment  of  S15, 000.00  at  the  end  of  the  ienth  year 
represented  a  balance  due  at  the  beginning  of  the  year 
plus  5%  interest  on  that  balance  accrued  during  the  year. 
In  other  words,  the  $15,000.00  is  105%  of  balance  due  at 
the  beginning  of  the  year. 

15,000  -5-  1.05  =  $14,285.71 

from  105,000  =  90,714.29 

the  amount  to  be  amortized  over  nine  years. 

To  find  the  amortization  factor: 

compound  amount  1.551328 

compound  interest  .551328 

divided  into  .05 

sinking  fund  factor  .0906901 

add  .05  .1406901 

multiply  by  90,714.29  $12,762.6025 

the  annual  amortization  of  principal  and  interest. 
To'  which  must  be  added  every  year  the  5  % 

interest  on  $14,285.71  714.2855 

Total  annual  cost  $13,476.8880 

But  if  the  problem  had  stated  the  present  worth  of  1 
for  nine  years  at  5%,  .644609,  it  would  have  been  a  simple 
matter  to  find  the  amortization  factor  directly: 
present  worth  .644609 

compound  discount  .355391 

divided  into  .05  .1406901  as  above 

In  general,  whenever  the  amount  of  1,  or  s  °,  is  given,  it 
is  easier  to  find  the  amount  of  an  annuity  or  the  sinking 
fund  factor;  whenever  the  present  worth  of  1,  or  v°, 
is  given,  it  is  easier  to  find  the  present  worth  of  an  annuity 
or  the  amortization  factor.  The  four  rules  are : 
compound  interest 


amount  of  annuity- 


single  interest 


162  MATHEMATICS  I  OR  THE  ACCOUNTANT 

.  ,  .       .      ,  ,    ^  single  interest 

sinking  fund  factor 

compound  interest 

^,      ,  .^     compound  discount 

present  worth  of  annuity — 

single  interest 

, .     , .       .    ,  single  interest 

Amortization  factor  — or 

compound  discount; 

amortization   factor  =  sinking   fund   factor   plus   single 

interest. 

Valuation  of  bonds:    The  handiest  formula  to  use  in 
bond  valuation  is  that  of  Mr.  Makeham.    It  requires  only 
the  finding  of  v  °,  the  present  worth  of  1.    It  is  as  follows: 
Let  C  be  the  redemption  value; 
0  the  coupon  rate; 
n  the  life  of  the  bond; 
i  the  desired  income  rate; 
K  the  present  worth  of  C  at  rate  i. 

The  value  of  this  bond  consists  of  the  present  worth  of 
the  redemption  value,  which  is  K,  and  the  present  worth 
of  the  coupons  at  the  desired  yield  rate  i.  If  the  yield 
rate  were  the  same  as  the  coupon  rate,  the  bond  would 
obviously  be  purchased  at  par.  The  excess  over  par, 
called  the  premium,  or  the  deficiency  under  par,  called 
the  discount,  depend  on  the  relation  existing  between 
these  rates.  If  the  purchaser  is  satisfied  with  a  yield  rate 
less  than  the  coupon  rate,  he  will  be  willing  to  pay  more 
than  par  for  the  bond.  If  he  desires  a  larger  yield  than 
the  coupon  rate,  he  will  purchase  at  a  discount.  The 
measure  of  this  premium  or  discount  is  the  ratio  of  coupon 
to  yield.  It  should  also  be  kept  in  mind  that  most  bonds 
pay  semiannual  coupons,  and  that  consequently  the 
annual  coupon  and  yield  rates  must  be  halved,  and  the 
time  must  be  doubled,  to  conform  with  the  half-year 
periods.  The  following  problem  will  illustrate  this: 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.     1G3 

Vahie  a  20-year  5%  bond,  redeemable  at  par  $1,000.00, 
on  a  4%  basis.    Given  that  v*"  at  2%  is  .45289. 
C  =  $1,000.00 

c  =  .025 

n  =  40 

i  =  .02 

K  =  1,000  X  .45289  =  452.89 

C-K  =  547.11 

If  the  bond  were  purchased  to  yield  5%,  that  is,  to  yield 

the  coupon  rate,  then  $547.11  would  be  the  present  worth 

of  the  coupons.    This  can  be  proved  by  computing  the 

present  worth  of  a  40-period  annuity  of  $25.00  at  23^%. 

But  the  yield  is  only  2%,  so  that  the  coupons  must  have 

cost 

not; 

^^^  of  $547.11  =  $    683.89 

.02 

adding  the  above  452.89 


cost  of  the  bond  $1,136.78 

Expressed  as  a  formula,  Makeham's  rule  is. 

Value  of  bond  =  Cv "  +— (C— K). 

1 

Rule :  Find  the  present  worth  of  the  redemption  value ; 

call  this  K.     Subtract  this  from  the  redemption  value, 

multiply  by  the  coupon  rate,  and  divide  by  the  income 

rate.    Add  this  last  result  to  K. 

Another  probleni  will  illustrate  this  for  a  bona  bought 

at  a  discount: 

Value  a  15-year  5%  bond  redeemable  at  $1,000.00  on  a 

6%  basis,    v^o  at  2^%  =  .476743. 

C  =  $1,000.00 

c  =  .025 

n  =  30 

i  =  .03 


164  MATHEMATICS  FOR  THE  ACCOUNTANT 

K  =  1,000  X  .476743  =  476.743 

C  —  K  =  523.257 

.025 


.03 


X  523.257     =  436.048 


value  $912,791 

Valuation  by  discounting:  Up  to  the  present  time  the 
Institute  problems  have  all  treated  short-term  bonds.  In 
such  cases  the  formula  is  not  necessary,  it  being  easier  to 
discount  from  maturity.  For  instance,  a  5%  bond, 
redeemable  at  1,000,  is  bought  on  a  4%  basis;  that  is, 
the  semiannual  yield  is  2%.  At  maturity  the  holder  will 
receive  par  $1,000.00  plus  coupon  $25.00.  So  at  the 
beginning  of  the  last  half-year  the  value  was  1,025  h-  1.02. 
At  that  time  a  coupon  was  payable  to  the  amount  of 
$25.00.  This  increased  the  value  of  the  bond  by  that 
amount.  Going  back  to  the  beginning  of  the  previous 
half-year,  the  value  is  found  by  discounting,  that  is, 
dividing  by  1.02;  and  another  coupon  again  increases  the 
value  of  the  bond.  Presented  in  a  vertical  schedule 
At  maturity,  par  $1,000.00 

plus  coupon  25.00 

$1,025.00 
value  6  months  before,  divide  by 

1.02  1,004.9019 

plus  coupon  25.00 

$1,029.9019 
value  6  months  before  that  date, 

divide  by  1.02  1,009.7078 

plus  coupon  25.00 

$1,034.7078 

Continue  until  the  date  stated  in  the  problem  has  been 
reached.  This  method  is  slightly  less  accurate  than  the 
valuation  by  formula,  because  of  the  error  in  the  right 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.    165 

hand  digits  caused  by  the  division.  The  method  is  useful 
only  in  cases  where  speed  is  necessary,  and  only  when  the 
term  of  the  bond  is  short. 

The  above  problem  occurred  on  the  Institute  examina- 
tion of  November,  1918.  The  redemption  value  of  the 
bond  in  that  problem  was  $10,000.00,  which  gives  the 
value  one  year  before  maturity  $10,347.0780 

value  1  Yi  years  before  maturity, 
divide  by  1.02  10,144.194 

if  bought  ex  interest ;  otherwise 
add  coupon  250.000 

Value  $10,394,194 

Redemption  at  a  premium  or  discount:  On  the  May, 
1919  examination  of  the  Institute  was  this  problem: 

A  bond  bearing  interest  at  5%  payable  annually  and 
repayable  in  5  years  with  a  bonus  of  10%  is  for  sale. 
What  price  can  a  purchaser  pay  who  desires  to  realize  6% 
on  his  investment?    v^  at  6%  is  .7473. 

The  noteworthy  feature  of  this  problem  is  that  the  bond 
is  redeemable  at  a  10%  premium;  that  is,  the  value  of 
C  in  Makeham's  formula  is  $1,100.00.  This  has  no  effect 
on  the  yield,  for  the  problem  expressly  states  that  the 
purchaser  desires  to  realize  6%  on  his  investment.  But 
the  coupon  rate  is  5%  on  the  face  of  the  bond,  regardless 
of  the  redemption  value.  And  $50.00  on  1,000  is  not  5% 
on  the  investment  as  purchased.     5%  on  1,000  is  only 

—  of  5%  when  adjusted  to  the  redemption  value  1,100. 

And  since  the  purchaser  bases  his  valuation  on  this 
redemption  value,  the  coupon  rate  must  also  be  based  on 
it.  We  can  value  this  bond  by  Makeham's  formula  more 
quickly  than  by  discounting: 


166  MATHEMATICS  FOR  THE  ACCOUNTANT 


G 

c                              = 

n                              = 

$1,100.00 

4  6/11% 
5 

i                               = 

K  =  1,100  X  .7473    = 
C  —  K  =  277.97 

6% 
822.03 

*  «/"  X  277.97         = 
6 

210.58 

Value 
The  solution  by  discounting  is: 
Value  at  maturity 
plus  coupon 

$1,032.61 

$1,100.00 
50.00 

$1,150.00 
value  one  year  before,  divide*  by  1.06  1,084.9057 

plus  coupon  50.00 

$1,134.9057 
value  two  years  before,  divide  by  1.06  1,070.6658 
plus  coupon  50.00 

$1,120.6658 
value  three  years  before,  divide  by  1.06  1,057.2319 
plus  coupon  50.00 

$1,107.2319 
value  four  yeaTs  before,  divide  by  1.06  1,044.5584 
plus  coupon  50.00 

$1,094.5584 
value  five  years  before,  divide  by  1.06        1,032.602 
if  purchased  ex  interest. 

Valuation  between  interest  dates  is  the  usual  situation 
in  actual  practise.  In  such  cases  the  value  as  of  the  last 
interest  date  must  be  first  determined.  The  change  in 
value  up  to  the  next  interest  date  consists  of  two  things: 
that  part  of  the  coupon  and  of  the  premium  or  discount 
proper  to  the  period.    The  difference  between  coupon  and 


PROBLEMS  FROxM  EXAMINATIONS  OF  THE  A.  I.  A.     167 

premium  or  discount  is  the  net  change  in  the  value  of  the 
bond.  If  the  bond  is  below  par  this  difference  causes  an 
increase  in  value ;  if  the  value  is  above  par,  this  difference 
causes  a  decrease  in  value.  The  simplest  method  is  to 
find  the  difference  between  coupon  and  discount  or  pre- 
mium and  add  it  to  the  value  of  the  bond  if  below  par, 
or  subtract  it  if  the  bond  is  above  par. 

If  the  bond  in  the  last  problem  had  been  purchased 
three  months  after  the  date  when  its  value  was  $1,032,602, 
the  value  would  have  been: 

Value  on  last  interest  date  $1,032,602 

1/4  of  6%  of  that  value  15.4890 
less  1/4  of  $50.00  12.50  2.989 

adding,  because  the  value  is  below 
redemption  value  $1,035,591 

Serial  bonds  have  never  been  required  in  Institute 
examinations.  In  fact,  the  valuation  of  a  series  is  usually 
a  complicated  process,  because  redemptions  are  usually 
at  a  premium  which  varies  as  the  final  date  approaches, 
and  the  amount  redeemed  at  the  various  dates  is  often 
variable.  If  a  series  is  regular  in  every  way  (that  is,  if  the 
redemptions  are  equal  in  amount,  interval,  and  redemp- 
tion value),  the  series  may  be  valued  by  Makeham's 
formula.  In  this  case  K  is  the  present  worth  of  a  series 
of  payments,  and  the  present  worth  of  an  annuity  must 
be  found  and  used  in  place  of  v  °. 

Problem:  Value  a  series  of  20  $1,000.00  bonds,  re- 
deemable at  par,  one  each  year,  beginning  one  year  from 
date,  on  a  4%  annual  basis.  The  bonds  pay  5%  annually. 
1.0420  =  2.1911231. 

We  must  find  the  present  worth  of  an  annuity: 
Compound  amount  2.1911231 

present  worth  .4563869 

single  interest  .04 

divided  into  .4563869  =  13.5903263 


1G8  MATHEMATICS  FOR  THE  ACCOUNTANT 

Each  redemption  is  $1,000.00;  the 
present   worth   is    1,000  X 

13.5903263  =  13,590.3263 

C— K  =  6,409.6737 

-  X  6,409.6737  =  8,012.0921 

4 


Value  $21,602.4184 

This  closes  the  discussion  from  the  investor's  standpoint. 
There  remains  one  phase  of  the  bond  problem  from  the 
standpoint  of  the  issuer:  the  writing  off  of  premium  or 
discount  at  which  the  bonds  were  originally  sold.  Such 
premium  or  discount  ought  to  be  amortized  or  accumu- 
lated on  the  annuity  principle,  but  many  bookkeepers  are 
not  competent  to  do  this  work,  and  the  preparation  of 
the  schedule  is  a  tiresome  task.  An  alternative  method, 
called  the  ''bonds  outstanding  method,"  was  required  in 
the  following  problem: 

The  A  company  issues  $100,000.00  bonds,  maturing  as 
follows : 

$  50,000.00  in  1  year 

75,000        "  2      « 

100,000.00  «  3      « 

150,000.00  "  4      « 

125,000.00  "  5      " 

These  bonds  were  issued  at  90.  State  the  amount  of 
discount  to  be  written  off  annually,  using  the  bonds  out- 
standing method. 

The  solution  in  the  Journal  of  Accountancy  opened 
with  this  quotation  from  Dickinson's  "Accounting  Prac- 
tise and  Procedure": 

"There  are  various  methods  in  use  for  determining  the 
proper  interest  charge  to  be  made  to  income  account 
under  the  varying  conditions  which  arise. 


PROBLEMS  FROM  EXAMINATIONS  OF  THE  A.  I.  A.    169 

"The  first  and  most  correct  method,  which  may  be 
called  the  effective  interest  method,  consists  in  charging 
to  income  account  the  effective  interest  calculated  from 
the  known  conditions  of  issue  upon  the  whole  amount 
outstanding  during  the  year. 

"The  second  and  more  common  method,  which  may  be 
called  the  equal  instalment  method,  is  to  ignore  altogether 
the  effective  interest  rate;  to  charge  to  income  account 
each  year  the  interest  actually  accrued,  together  with  a 
proportionate  part,  according  to  the  term  of  issue,  of 
the  discount  on  issue  or  premium  on  redemption. 

"A  third  method  which  may  be  called  the  bonds  out- 
standing method,  which  may  be  safely  adopted  where 
by  reason  of  complication  in  the  terms  of  issue  or  redemp- 
tion it  is  difficult  or  impracticable  to  determine  the  true 
interest  rate,  is  to  distribute  the  discount  or  premium 
over  the  period  in  the  proportion  that  the  bonds  out- 
standing for  each  year  bear  to  the  sum  of  the  bonds 
outstanding  for  all  the  years  of  the  currency  of  the 
loan. 

"When  a  large  accumulated  surplus  is  available  the 
practise  is.  irequently  adopted  of  charging  the  whole 
discount  on  issue  to  profit  and  loss  account.  A  great 
objection  to  this  practise  is  that  thereby  the  true  rate 
of  interest  on  loans  during  their  currency  is  entirely  lost 
sight  of;  current  fixed  charges  are  understated;  and  the 
discount  is  charged  against  surplus  arising  out  of  previous 
operations  instead  of  against  income  from  current  opera- 
tions which  should  meet  it. 

"The  charge  to  income  account  by  bonds  outstanding 
method  is  so  close  to  that  given  by  the  effective  interest 
method  that  for  all  practical  purposes  it  may  safely  be 
adopted." 


170  MATHEMATICS  FOR  THE  ACCOUNTANT 

Table  Showing  Bonds  Outstanding  Each  Year,  and  Pro- 
portion to  Total  Outstanding  during  the  Entire  Term 
of  the  Loan,  and  Corresponding  Reduction  of 
Discount 


Par  of  Bonds 

Proportion 

Reduction  of 

Year 

Outstanding 

of  Total 

Discount 

1 

$    500,000.00 

20/69 

$14,492.75 

2 

450,000.00 

18/69 

13,043.48 

3 

375,000.00 

15/69 

10,869.57 

4 

275,000.00 

11/69 

7,971.01 

5 

125,000.00 

5/69 

3,623.19 

Totals 

$1,725,000.00 

69/69 

$50,000.00 

Since  the  bonds  were  issued  at  90  there  w^as  a  discount 
of  10%  of  $500,000.00,  or  $50,000.00.  During  the  first 
year  the  bonds  outstanding  amounted  to  $500,000.00;  at 
the  end  of  that  year  $50,000.00  were  redeemed,  leaving 
$450,000.00  outstanding  during  the  second  year,  and  so 
on.      The    footing    of    this     Outstanding    column    is 

$1,725,000.00.    Of  this  amount      ^^'^      or  -  were  out- 
'       '  1,725,000       69 

standing  during  the  first  year,  and  so  that  year  must  bear 

20 

—of  the  total  discount,  $50,000.00,  or  $14,492.75.    The 
69 

remainder  of  the  schedule  needs  no  explanation.    Of  course 

the  various  columns  must  prove :  the  total  of  the  fractions 

69' 
must  be  —   and  the  total  of  the  Reduction  of  Discount 
69 

column  must  be  $50,000.00. 


APPENDIX 

LOGARITHMIC  TABLES 

Table  I — The  Compound  Amount  of  One  Unit. 

Table  II— The  Present  Worth  of  One  Unit. 

Tal)le  III — The  Amount  of  Periodic  Payments  of  One 
Unit  Each. 

Table  IV — The  Present  Worth  of  Periodic  Payments 
of  One  Unit  Each. 

Table  V— The  Periodic  Payment  That  Will  Amount  to 
One.    The  Sinking  Fund. 

Table  VI— Effective  Rate  Factors. 

Table  VII — Ten-Place    Logarithms    of    the    Interest 
Ratios. 


CONSTEUCTION  OF  TABLES 

The  following  brief  description  of  the  manner  in  which 
the  tables  in  this  book  have  been  constructed  will  be  based 
on  the  previous  sections  of  the  book  in  which  the  various 
functions  of  s"  were  discussed.  There  is  nothing  difficult 
or  interesting  in  the  construction  of  tables,  and  in  fact  only 
their  usefulness  can  justify  the  manual  labor  required  in 
their  construction. 

It  has  seemed  desirable  to  include  all  the  values  of  the 
various  factors  for  one  to  one  hundred  periods,  and  to  eight 
places  of  decimals.  If  more  than  a  hundred  periods  are 
specified  in  any  problem,  the  proper  value  of  s"  or  v"  can 
be  found  by  multiplying  two  or  more  table  values ;  if  a  value 
of  some  annuity  factor  is  required,  it  can  then  be  derived 
from  this  value  of  s"  or  v"  by  formula.  If  more  than  eight 
places  of  decimals  are  necessary  for  accuracy,  it  will  be 
necessary  either  to  build  a  schedule  covering  the  life  of  the 
transaction,  and  adjust  errors,  or  it  will  be  necessary  to 
find  the  required  factor  by  the  use  of  twelve  place  loga- 
rithms, as  will  be  explained  below. 


Table  I:  The  Compound  Amount  of  One  Unit. 
Great  care  must  be  used  in  constructing  this  table,  as  it  is 
the  basis  of  two  of  the  other  tables,  and  any  error  in  this 
table  will  be  especially  magnified  in  the  table  of  s„.  The 
table  is  developed  by  ordinary  multiplication,  the  decimals 
running  to  not  less  than  twelve  places,  and  fourteen  are 
better.  Every  ten  periods  at  most  there  should  be  an  in- 
dependent check,  either  by  use  of  some  machine,  or,  as  in 
this  case,  by  use  of  the  fifteen-place  logs  of  1+i  given  in  the 
Si^rague-Perrine  "Accountancy  of  Investment",  the  anti- 
logs  being  found  by  use  of  the  tables  of  factors  in  the  same 
book.  If  any  error  in  the  value  as  given  by  actual  multi- 
plication is  found,  the  value,  and  some  preceding  values 
should  be  adjusted  before  proceeding  with  further  multi- 
plication. As  an  instance,  the  table  of  s"  at  5%,  beginning 
at  50,  is  made  up  as  follows : 

50  11.467399785753 

.573369989288 


51 

12.040769775041 
.602038488762 

52 

12.642808263793 
.632140413190 

53 

13.274948676983 
.663747433849 

54 

13.9.386961108.32 
.696934805542 

55 

14.635630916374 
.731781545819 

56 

15.367412462193 
.768370623110 

57 

16.1.35783085303 
.806789154265 

58 

16.942572239568 
.847128611978 

59 

17.789700851546 
.889485042577 

60  18.679185894123 

Xow  the  logarithmic  value  should  be  very  carefully  cal- 
culated, and  if  any  discrepancy  is  discovered  which  affects 
any  figure  except  the  two  at  the  right,  a  suitable  adjust- 
ment should  be  made.  Such  adjustment  may  be  on  a  pro- 
portional basis;  that  is,  if  the  error  were  16,*  add  16  to  the 
last  value,  14  to  the  preceding  one,  and  so  on  proceeding 
liackward  as  far  as  desired. 


Table  II:     The  Present  Worth  of  One  Unit. 

This  table  is  formed  by  multiplication,  but  is  peculiar 
in  that  the  actual  work  begins  with  the  value  for  100  peri- 
ods. Of  course  we  might  begin  with  one  period,  dividing 
1  by  1+i,  dividing  this  result  by  1+i,  and  so  on.  But  the 
multiplication  is  easier,  and  gives  the  same  results.  The 
logarithmic  value  for  100  periods  is  calculated,  and  multi- 
plication proceeds  until  the  value  for  90  periods  is  reached. 
This  value  is  checked  by  logarithms,  and  the  multiplication 
is  resumed.     The  table  for  5%,  beginning  at  60  periods  is: 


60 

.053535523746 
.002676776187 

59 

.056212299933 
.002810614996 

58 

.059022914929 
.002951145746 

57 

.061974060675 
.003098703033 

56 

.065072763708 
.003253638185 

55 

.068326401893 
.003416320094 

54 

.071742721987 
.003587136099 

53 

.075329858086 
.003766492904 

52 

.079096350990 
.003954817549 

51 

.083051168539 
.004152558427 

50 

.087203726966 

The  logarithmic  value  for  50  periods  must  now  be  cal- 
culated, and  any  necessary  adjustment  made  before  pro- 
ceeding with  the  multiplication. 


Table  III :  The  Amount  of  Periodic  Payments  of 
One  Unit  Each. 

As  was  shown  on  page  61,  such  amounts  may  be  found 
by  addition  from  values  of  the  compound  amount.  This 
table  has  been  so  constructed,  and  checked  every  ten  peri- 

ods  by  the  formula  Sj,= — -. — .     For  instance,  the  value  of 

s"  at  5%  for  50  periods  has  just  been  given  as 
11.467399785753.  Deducting  1  and  dividing  by  .05  we  have 
s'»=209.34799571506.  To  find  s'^  for  51  periods  we  must 
add  to  this  value  s"  for  50  periods.  The  reason  for  this  is 
evident.  In  any  annuity,  the  payments  are  made  at  the 
end  of  the  period.  A  payment  therefore  does  not  accumu- 
late interest  until  the  end  of  the  follomng  period.  The 
payment  made  at  the  end  of  the  first  period  mil  have  ac- 
cumulated, at  the  end  of  the  fifty-first  period,  only  fifty 
periods  interest.     The  schedule  is  built  as  follows: 


50 

209.34799571506 
11.467399785753 

51 

220.815395500813 
12.040769775041 

52 

232.856165275854 
12.642808263793 

53 

245.498973539647 
13.274948676983 

54 

258.773922216630 

and  so  on.  At  60  periods  the  value  should  be  calculated 
from  the  value  of  s"  for  60  periods,  and  any  discrepancy 
should  be  adjusted  over  as  many  periods  as  may  seem 
necessary. 


Tablk  IV :     The  Present  Worth  of  Periodic  Payments 

OF  One  Unit  Each. 

This  table  is  formed  by  addition  from  the  table  of  v" 

in  precisely  the  same  manner  as  Table  III  is  formed  from 

Table  I.     The  checks  are  made  from  the  corresponding 

1 yn 

value  of  V"  according  to  the  formula  a^=    . —  .     The  value 

of  V"  for  50  periods  being  .087203726966,  we  deduct  from  1 
and  divide  by  .05,  and  the  result  is  a^  for  50  periods, 
18.25592546068.  The  table  proceeds  by  addition  until  the 
value  for  60  periods  is  reached.  This  value  should  be 
checked  and  adjusted  in  a  mananer  similar  to  that  de- 
scribed for  the  table  of  s^. 
Table  V :     The  Periodic  Payment  That  Will  Amount 

TO  One. 

The  Sinking  Fund :  This  table  is  formed  by  dividing 
the  successive  values  of  s^  into  1.  There  is  no  method  of 
checking  such  division  except  by  differencing.  This 
method,  as  applied  to  bond  schedules,  was  described  on 
pages  104  and  105.  The  third  order  of  differences  should 
be  sufficient  to  show  whether  the  schedule  is  accurate.  Any 
considerable  variation,  particularly  in  case  a  difference 
should  be  larger  than  the  preceding  one,  will  indicate  an 
error  in  the  schedule. 

The  periodic  payment  that  Avill  extinguish  a  debt  of 
one  unit — the  amortization  factor — is  often  given  in  place 
of  the  sinking  fund  factor.  But  the  schedule  presented  on 
page  95  should  make  it  clear  that  the  amortization  factor 
can  always  be  found  by  adding  i  to  the  sinking  fund  factor. 
It  is  desirable  to  give  the  sinking  fund  factors  because  they 
are  more  often  used  in  this  book,  especially  in  the  various 
problems  dealing  with  the  valuation  of  assets. 

Table  VI :     The  Effective  Rate  Factor. 

This  quantity  was  discussed  at  length  on  pages  51,  52, 
and  54.  The  process  of  finding  the  values  is  logarithmic, 
and  the  antilogs  ma}^  be  found  from  the  table  of  factors,  or 
less  accurately  from  the  ten-place  logs  of  1+i  at  the  end 
of  this  book. 

The  effective  rate  factor  for  use  with  annuity  tables 
(see  pages  63  to  65)  is  not  given  because  it  is  seldom  re- 
quired. If  it  is  necessary  to  change  from  a  table  value  at 
an  annual  rate  to  an  effective  rate,  the  rule  is : 

^  In  case  of  Sn  or  a^,  multiply  the  table  value  by  i,  and 
divide  by  the  correct  value  of  j  ^ . 

In  case  of  „  or  —  mult i pi v  by  i   and  divide  by  L 


The  Compound  Amount  of  One  Unit. 
s"  =  (l+i)" 


n 

1% 

iy4% 

1M.'% 

1%% 

2% 

1 

1.01 

1.0125 

1.015 

1.0175 

1.02 

2 

1.0201 

1.02515625 

1.030225 

1.035:30625 

1.0404 

3 

1.030301 

1.03797070 

1.045678.38 

1.05:342411 

1.061208 

4 

1.04060401 

1.05095434 

1.06136355 

1.0718.5903 

1.08243216 

5 

1.05101005 

1.06408215 

1.07728400 

1.09061656 

1.10408080 

6 

1.06152015 

1.07738318 

1.09344326 

1.10970235 

1.12616242 

7 

1.07213535 

1.09085047 

1.01984491 

1.12912215 

1.14868567 

8 

1.08285671 

1.10448610 

1.12649259 

1.14888178 

1.17165938 

9 

1.09368527 

1.11829218 

1.14338998 

1.16898721 

1.19.509257 

10 

1.10462213 

1.13227083 

1.16054083 

1.18944449 

1.21899442 

11 

1.11566835 

1.14642422 

1.17794894 

1.21025977 

1.24337431 

12 

1.12682503 

1.16075452 

1.19561817 

1.2314:3931 

1.26824179 

13 

1.13809328 

1.17526395 

1.21355244 

1.25298950 

1.29360663 

14 

1.14947421 

1.18995475 

1.23175573 

1.27491682 

1.31947876 

15 

1.1609G896 

1.20482918 

1.25023207 

1.29722786 

1.34586834 

16 

1.17257864 

1.21988955 

1.26898555 

1.31992935 

1.37278571 

17 

1.1 84:30443 

1.23513817 

1.28802033 

1.34302811 

1.40024142 

18 

1.19614748 

1.25057739 

1.30734064 

1.36653111 

1.42824625 

19 

1.20810895 

1.26620961 

1.32695075 

1.39044540 

1.45681117 

20 

1.22019004 

1.28203723 

1.34685.501 

1.41477820 

1.48594740 

21 

1.23239194 

1.29806270 

1.36705783 

1.43953681 

1.51566634 

22 

1.24471586 

1.31428848 

1.38756370 

1.46472871 

1.54597967 

23 

1.25716302 

1.33071709 

1.40837715 

1.49036146 

1.57689926 

24 

1.26973465 

1.34735105 

1.42950281 

1.51644279 

1.60843725 

25 

1.28243200 

1.36419294 

1.45094535 

1.54298054 

1.64060599 

26 

1.29525631 

1.38124535 

1.47270953 

1.56998260 

1.67341811 

27 

1.30820888 

1.39851092 

1.49480018 

1.59745739 

1.70688648 

28 

1.32129097 

1.41599230 

1.51722218 

1.62541290 

1.74102421 

29 

1.33450388 

1.43369221 

1.53998051 

1.65385762 

1.77584469 

30 

1.34784892 

1.45161336 

1.56308022 

1.68280013 

1.81136158 

31 

1.361.32740 

1.46975853 

1.58652642 

1.71224913 

1.84758882 

32 

1.37494068 

1.48813051 

1.610324:32 

1.74221:349 

1.88454059 

33 

1.38K6!)009 

1.50673214 

1.63447919 

1.77270223 

1.92223140 

34 

1.40257699 

1.52556629 

1.65899637 

1.80372452 

1.96067603 

35 

1.4166027G 

1.54463587 

1.68388132 

1.83528970 

1.99988955 

36 

1. -43076878 

1.56394382 

1.70913954 

1.86740727 

2.03988734 

37 

1.44507647 

1.58349312 

1.73477663 

1.90008689 

2.08068509 

38 

1.45952724 

1.60328678 

1.76079828 

1.9:33:33841 

2.12229879 

39 

1.47412251 

1.62332787 

1.78721025 

1.96717184 

2.16474477 

40 

1.48886373 

1.64361946 

1.81401841 

2.00159734 

2.20803966 

41 

1.50375237 

1.66416471 

1.84122868 

2.03662530 

2.25220046 

42 

1.51878989 

1.68496677 

1.86884712 

2.07226624 

2.29724447 

43 

.1.53397779 

1 .7()()02885 

1 .89687982 

2.1085:3090 

2.34318936 

44 

1.54931757 

1.72735421 

1.9253:3302 

2.1454:3019 

2.3900.5314 

45 

1.56481075 

1.74894614 

1.95421301 

2.18297522 

2.4:J785421 

n 

1% 

iy4% 

IV2% 

1%% 

2% 

46 

1.58045885 

1.77080797 

1.98352621 

2.22117728 

2.48661129 

47 

l.5iHi2(iSU 

1.79294:306 

2.01327910 

2.260047H9 

2.53634351 

48 

I.(jr222(i08 

1.81.535485 

2.04347829 

2.29959H72 

2.5870703!) 

49 

1.628:31^3-1. 

1.83804679 

2.0741:3046 

2.:3:3984170 

2.6:3881179 

50 

l.()H.(i;5182 

1.86102237 

2.10524242 

2.3807889:3 

2.69158803 

51 

l.()()0178M. 

1.88428515 

2.13682106 

2.42246274 

2.74541979 

52 

l.«77()8892 

1.9078:3872 

2.16887337 

2.46484566 

2.800:32819 

53 

l.()914()581 

1.93168670 

2.20140647 

2.50798046 

2.85633475 

54 

1.711U04.7 

1.95583279 

2.23442757 

2.55187012 

2.91346144 

55 

1.72852+57 

1.98028070 

2.26794398 

2.59652785 

2.97173067 

56 

1.71-580982 

2.00503420 

2.30196314 

2.64196708 

3.03116529 

57 

1.7();}2()792 

2.0:3009713 

2.33649259 

2.68820151 

:}.091 78859 

58 

1.78090060 

2.05547335 

2.37153998 

2.73524503 

3.15362436 

59 

1.79870960 

2.08116676 

2.40711308 

2.78311182 

3.21669685 

60 

1.81669670 

2.10718135 

2.44321978 

2.83181628 

3.28103079 

61 

1.83486367 

2.13352111 

2.47986807 

2.88137306 

3.34665140 

62 

1.85321230 

2.16019013 

2.51706609 

2.93179709 

8.4135844.3 

63 

1.87174443 

2.18719250 

2.55482208 

2.98310354 

3.48185612 

64 

1.89046187 

2.2145:3241 

2.59314442 

3.03530785 

3.55149:324 

65 

1.90936649 

2.24221407 

2.63204158 

3.08842574 

3.62252311 

66 

1.92846015 

2.27024174 

2.67152221 

3.14247319 

3.69497357 

67 

1.94774475 

2.29861976 

2.71159504 

3.19746647 

3.76887:304 

68 

1.96722220 

2.32735251 

2.75226896 

3.25342213 

3.84425050 

69 

1.98689442 

2.35644442 

2.79355300 

3.31035702 

3.92113551 

70 

2.00676337 

2.38589997 

2.83545629 

3.36828827 

3.99955822 

71 

2.02683100 

2.41572372 

2.87798814 

3.42723331 

407954939 

72 

2.04709931 

2.44592027 

2.92115796 

3.48720990 

4.16114037 

73 

2.06757031 

2.47649427 

2.96497533 

3.54823607 

4.24436318 

74 

2.08824601 

2.50745045 

3.00944996 

3.61033020 

4.32925045 

75 

2.10912849 

2.53879358 

3.05459171 

3.67351098 

4.41583556 

76 

2.13021975 

2.57052845 

3.10041058 

8.73779742 

4.50415126 

77 

2.15152195 

2.60266011 

3.14691674 

8.80320888 

459423521 

78 

2.17303717 

2.6:3519336 

3.19412049 

3.86976503 

4.68611991 

79 

2.19476754 

2.66813328 

3.24203230 

3.93748592 

4.77984231 

80 

2.21671522 

2.70148494 

3.29066279 

4.00639192 

487543916 

81 

2.23888237 

2.73525350 

3.34002273 

4.07650378 

4.97294794 

82 

2.26127119 

2.76944417 

3.39012307 

4.14784260 

5.07240690 

83 

2.28388390 

2.80406222 

3.44097492 

4.22042984 

5.17:385504 

84 

2.30672274 

2.83911300 

3.49258954 

4.29428737 

5.27733214 

85 

2.32978997 

2.87460191 

3.54497838 

436943740 

5.38287878 

86 

2.85308787 

2.91053444 

3.59815.306 

444590255 

5.49053636 

87 

2.37661875 

2.94691612 

3.65212535 

452370584 

5.60034708 

88 

2.40038494 

2.98375257 

3.70690723 

4.60287070 

5.712:35402 

89 

2.42438879 

3.02104948 

3.76251084 

4.68342093 

5.82660110 

90 

2.44863267 

3.05881260 

3.81894851 

4.76538080 

5.94313313 

91 

2.47311900 

3.09704775 

3.87623273 

4.84877496 

6.06199579 

92 

2.49785019 

3.13576085 

3.93437622 

493362853 

6.18:323570 

93 

2.52282869 

3.17495786 

3.993.39187 

5.01996703 

6.30690042 

94 

2.54805698 

3.21464483 

4.05329275 

5.10781645 

6.43303843 

95 

2.57353755 

3.25482789 

4.11409214 

5.19720324 

6.56169920 

96 

2.59927293 

3.29551324 

4.17580352 

5.28815429 

6.6929:3318 

97 

2.62526565 

3.33670716 

423844057 

5.38069699 

6.82679184 

98 

2.65151831 

3.37841600 

4.:30201718 

5.47485919 

6.963:32768 

99 

2.67803349 

3.42064620 

4.36654744 

6.57066923 

7.10259423 

100 

2.70481383 

3,46340427 

4.43204566 

6.66815694 

7.24464612 

n 

21/4  % 

2y2% 

2y4% 

3% 

31/2% 

1 

1.0225 

1.025 

1.0275 

1.03 

1.035 

2 

1.04550625 

1.050625 

1.05575625 

1.0609 

1.071225 

3 

1. 060930 U 

1.07689063 

1.08478955 

1.092727 

1.10871788 

4 

1.09308332 

1.01381289 

1.11462126 

1.12550881 

1.14752300 

5 

1.11767769 

1.13140821 

1.14527334 

1.15927407 

1.18768631 

6 

1.142825H. 

1.15969342 

1.17676836 

1.19405230 

1.22925533 

7 

1.1685390 1 

1.18868575 

1.20912949 

1.22987387 

1.27227926 

8 

1.191-83114. 

1.21840290 

1.24238055 

1.26677008 

1.31680904 

9 

1.22171484 

1.24886297 

1.27654602 

1.30477318 

1.36289735 

10 

1.24920348 

1.28008454 

1.31165103 

1.34391638 

1.41059876 

11 

1.27731050 

1.31208666 

1.34772144 

1.38423387 

1.45996972 

12 

1.30604999 

1.34488882 

1.38478378 

1.42576089 

1.51006866 

13 

1.3354:3611 

1.37851104 

1.42286533 

1.46853371 

1.56395606 

14 

1.36548343 

1.41297382 

1.46199413 

1.51258972 

1.61869452 

15 

1.39620680 

1.44829817 

1.50219896 

1.55796742 

1.67534883 

16 

1.42762146 

1.48450562 

1.54350944 

1.60470644 

1.73398604 

17 

1.45974294 

1.52161826 

1.58595595 

1.65284763 

1.79467555 

18 

1.49258716 

1.55965872 

1.62956973 

1.70243306 

1.85748120 

19 

1.52617037 

1.59865019 

1.67438290 

1.75350605 

1.92250312 

20 

1.56050920 

1.63861644 

1.72042843 

1.80611123 

1.98978887 

21 

1.59562066 

1.67958185 

1.76774021 

1.86029457 

2.05943148 

22 

1.63152212 

1.72157140 

1.81635307 

1.91610341 

2.13151158 

23 

1.66823137 

1.76461068 

1.86630278 

1.97358651 

2.20611148 

24 

1.70576658 

1.80872595 

1.91762610 

2.03279411 

2.28332849 

25 

1.74414632 

1.85394410 

1.97036082 

2.09377793 

2.36324498 

26 

1.78338962 

1.90029270 

2.02454575 

2.15659127 

2.44595856 

27 

1.82351588 

1 .94780002 

2.08022075 

2.22128901 

2.53156711 

28 

1.86454499 

1.99649502 

2.13742682 

2.28792768 

2.62017196 

29 

1.90649725 

2.04640739 

2.19620606 

2.35656551 

2.71187798 

30 

1.94939344 

2.09756758 

2.25660173 

2.42726247 

2.80679370 

31 

1.99325479 

2.15000677 

2.31865828 

2.50008035 

2.90503148 

32 

2.03810303 

2.20375694 

2.38242138 

2.57508276 

3.00670759 

33 

2.08396034 

2.25885086 

2.44793797 

2.65233523 

3.11194235 

34 

2.13084945 

2.31532213 

2.51525626 

2.73190530 

3.22086033 

35 

2.17879356 

2.37320519 

2.58442581 

2.81386245 

3.33359045 

36 

2.22781642 

2.43253532 

2.65549752 

2.89827833 

3.45026611 

37 

2.27794229 

2.49334870 

2.72852370 

2.98522668 

3.57102543 

38 

2.32919599 

2.55568242 

2.80355810 

3.07478348 

3.69601132 

39 

2.381 G0290 

2.61957448 

2.88065595 

3.16702698 

3.82537171 

40 

2.43518897 

2.68506384 

2.95987399 

3.26203779 

3.95925972 

41 

2.48998072 

2.75219043 

3.04127052 

3.35989893 

4.09783381 

42 

2.54600528 

2.82099520 

3.12490546 

3.46069589 

4.24125799 

43 

2.60329040 

2.89152008 

3.21084036 

3.56451677 

4.38970202 

44 

2.44186444 

2.86380808 

3.29913847 

3.67145227 

4.54334160 

45 

2.72175639 

3.03790328 

3.38986478 

3.78159584 

4.70235855 

46 

2.78299590 

3.11385086 

3.48308606 

3.89504372 

4.86694110 

47 

2.84561331 

3.19169713 

3.57887093 

4.01189503 

5.03728404 

48 

2.90963961 

3.27148956 

3.67728988 

4.13225188 

5.21358898 

49 

2.97510650 

3.35327679 

3.77841535 

4.25621944 

5.39606459 

50 

3.04204640 

3.43710872 

3.88232177 

4.38390602 

5.58492686 

n 

2 1/4  7c 

21/2% 

2%% 

3% 

3y2% 

51 

3.1 10 1.92 14 

3.52303644 

3.98908562 

4.51542320 

5.780.399.30 

52 

3.1801.7852 

8.611112.35 

4.09878547 

4.65008590 

5.98271327 

53 

3.2520:5929 

3.701.39016 

4.211.50208 

4.79041247 

6.19210824 

54 

3.32521017 

3.79392491 

4.32731838 

4.93412485 

6.40883202 

55 

3.4-0002740 

3.88877303 

4.44631964 

5.08214859 

6.63;il4114 

56 

3.4.7652802 

3.98599236 

4.56859343 

5.23461305 

6.86.530108 

57 

3.55474990 

4.09564217 

4.69422975 

5..39165144 

7.01558662 

58 

3.63473177 

4.18778.322 

4.82.3.32107 

5.55.340098 

7.-354282 15 

59 

3.71651324 

4.29247780 

4.95596239 

5.72000.301 

7.61168203 

60 

3.80013479 

4.39978975 

5.09225136 

5.89160310 

7.87809090 

61 

3.88563782 

4.50978449 

5.23228827 

6.06835120 

8.15-382408 

62 

3.97306467 

4.62252910 

5.37617620 

6.25040173 

8.43920793 

63 

4.06245862 

4.73809233 

5.52402105 

6.43791379 

8.73458020 

64 

4.15386394 

4.85654464 

5.67593162 

6.63105120 

9.04029051 

65 

4.24732588 

4.97795826 

5.8.3201974 

6.82998273 

9.35670068 

66 

4.34289071 

5.10240721 

5.99240029 

7.03488222 

9.68418.520 

67 

4.44060576 

5.22996739 

6.157191.30 

7.24592868 

10.02313168 

68 

4.54051939 

5.360716.58 

6..32651406 

7.46.330654 

10.37-394129 

69 

4.64268107 

5.49473449 

6.50049319 

7.68720574 

10.73702924 

70 

4.74714140 

5.63210286 

6.67925676 

7.91782191 

11.11282526 

71 

4.85395208 

5.77290543 

6.86293632 

8.15535657 

11.-50177414 

72 

4.96316560  r 

5.91722806 

7.05166706 

8.40001727 

11.9043.3624 

73 

5.07483723 

6.06515877 

7.24558791 

8.65201778 

12.-32098801 

74 

5.18902107 

6.21678774 

7.44484158 

8.91157832 

12.7-5222259 

75 

5.30577405 

6.37220743 

7.64957472 

9.17892567 

13.19855038 

76 

5.42515396 

6.53151261 

7.85993802 

9.45429344 

13.66049964 

77 

5.54721993 

6.69480043 

8.07608632 

9.73792224 

14.1-3861718 

78 

5.67203237 

6.86217044 

8.29817869 

10.03005991 

14.6-3346873 

79 

5.79965310 

7.03372470 

8.52637861 

10.3.3096171 

15.14564014 

80 

5.93014.530 

7.20956782 

8.76085402 

10.64089056 

15.67573754 

81 

6.06.357357 

7.28980701 

9.00177751 

10.96011727 

16.224388-35 

82 

6.20000397 

7.5745.5219 

9.24932639 

11.28892079 

16.79224195 

83 

6.33950406 

7.76391599 

9.50368286 

11.62758842 

17.37997041 

84 

6.48214920 

7.95801389 

9.7650-3414 

11.97641607 

17.988269-38 

85 

6.62799112 

8.15696424 

10.0.3357258 

12.33570855 

18.61785881 

86 

6.77712092 

8.36088834 

10.30949583 

12.70577981 

19.26948387 

87 

6.92960614 

8.56991055 

10.59300696 

13.08695320 

19.94391.580 

88 

7.08552228 

8.7841.5832 

10.88431465 

13.47956180 

20.6419-5285 

89 

7.24494653 

9.00376227 

11.18363331 

13.88394865 

21.36442120 

SO 

7.40795782 

9.22885633 

11.49118322 

14.30046711 

22.11217595 

91 

7.57463688 

9.45957774 

11.80719076 

14.72948112 

22.88610210 

92 

7.74506621 

9.69606718 

12.13188851 

15.17136556 

23.68711568 

93 

7.919.33020 

9.93846886 

12.46551544 

15.62650652 

24.51616478 

94 

8.09751512 

10.1869.3058 

12.80831711 

16.09530172 

25.37423049 

95 

8.27970921 

10.44160385 

13.16054584 

16.57816077 

26.26232856 

96 

8.46600267 

10.70264.395 

13.52246085 

17.07550559 

27.18151006 

97 

8.65648773 

10.97021004 

13.89432852 

17.58777076 

28.13286291 

98 

8.8512.5871 

11.24446.5.30 

14.27642255 

18.11540388 

29.11751311 

99 

9.05041203 

11.52.557693 

14.66902417 

18.65886600 

30.13662607 

100 

9.25404630 

11.81371635 

15.07242234 

19.21863198 

31.19140798 

n 

'1% 

*y2% 

5% 

6y2% 

6% 

1 

1.04 

1.045 

1.05 

1.055 

1.06 

2 

1.0816 

1.092025 

1.1025 

1.113025 

1.1236 

3 

1.124^fi4. 

1.14116613 

1.157625 

1.17424138 

1.191016 

4 

1.1 69858.56 

1.19251860 

1.21550625 

1.23882465 

1.26247696 

5 

1.21665290 

1.24618194 

1.27628156 

1.30696001 

1.33822558 

6 

1.26531902 

1.30226012 

1.. 34009564 

1.37884281 

1.41851911 

7 

1.31593178 

1.36086183 

1.40710042 

1.45467916 

1.60363026 

8 

1.36856905 

1.42210061 

1.47745544 

1.53468651 

1.59384807 

9 

1.42331181 

1.48609514 

1.551.32822 

1.61909427 

1.68947896 

10 

1.48024428 

1.55296942 

1.62889463 

1.70814446 

1.79084770 

11 

1.53945406 

1.62285305 

1.71033936 

1.80209240 

1.89829856 

12 

1.6010.3222 

1.69588143 

1.79585633 

1.90120749 

2.01219647 

13 

1.66507351 

1.77219610 

1.88564914 

2.00577390 

2.13292826 

14 

1.73167645 

1.85194491 

1.97993160 

2.11609146 

2.26090396 

15 

1.80094361 

1.93528244 

2.07892818 

2.23247649 

2.39655819 

16 

1.87298125 

2.02237015 

2.18287459 

2.35526270 

2.54036168 

17 

1.94790050 

2.113.37681 

2.29201832 

2.48480215 

2.69277279 

18 

2.02581652 

2.20847877 

2.40661923 

2.62146627 

2.85433915 

19 

2.10684918 

2.30786031 

2.52695020 

2.76564691 

3.02569960 

20 

2.19112314 

2.41171402 

2.65329771 

2.91775749 

3.20713647 

21 

2.27876807 

2.52024116 

2.78596259 

3.07823415 

3.39956360 

22 

2.36991879 

2.63365201 

2.92526072 

8.24753703 

3.60353742 

23 

2.46471554 

2.75216635 

3.07152376 

3.42615157 

3.81974966 

24 

2.563.30416 

2.87601.383 

3.22509994 

3.61458990 

4.04893464 

25 

2.66583633 

2.00543446 

3.38635494 

3.81339235 

4.29187072 

26 

2.77246978 

3.14067901 

3.55567269 

4.02312893 

4.54938296 

27 

2.88336858 

3.28200956 

3.73345632 

4.24440102 

4.82234694 

28 

2.99870332 

3.42969999 

3.92012914 

4.47784307 

5.11168670 

29 

3.11865145 

3.58403649 

4.1161.3560 

4.72412444 

6.41838790 

30 

3.2433975 | 

3.74531813 

4.32194238 

4.98395129 

6.74349117 

31 

3.37313341 

3.91385745 

4.53803949 

5.25806861 

6.08810064 

32 

3.50805875 

4.08998104 

4.76494147 

5.54726238 

6.45338668 

33 

3.64838110 

4.27403018 

5.00318854 

5.85236181 

6.84068988 

34 

3.79431634 

4.46636154 

5.25334797 

6.17424171 

7.26102528 

35 

3.94608899 

4.66734781 

5.51601537 

6.51382501 

7.68608679 

36 

4.10393255 

4.87737846 

5.79181614 

6.87208538 

8.14725200 

37 

4.26808986 

5.09686049 

6.08140694 

7.25005008 

8.63608712 

38 

4.4.3881345 

5.32621921 

6.38547729 

7.64880283 

9.15425235 

39 

4.61636599 

5.56589908 

6.70475115 

8.0G948()99 

9.70350749 

40 

4.80102063 

5.81636454 

7.03998871 

8.51330877 

10.28571794 

41 

4.99306145 

6.07810094 

7.39198815 

8.98154076 

10.90286101 

42 

5.19278391 

6.35161548 

7.76158756 

9.475,52550 

11.55703267 

43 

5.40049527 

6.6374.3818 

8.14966693 

9.99667940 

12.25046463 

44 

5.61651508 

6.93612290 

8.55716028 

10.54649677 

12.98548191 

45 

5.84117568 

7.24824843 

8.98500779 

11.12655409 

13.76461083 

46 

6.07482271 

7.57441961 

9.43425818 

11.73851456 

14.69048748 

47 

6.31781.562 

7.91526849 

9.90597109 

12.38413287 

15.46591673 

48 

6.57052824 

8.2714.5-557 

10.40126965 

13.06526017 

16.39387173 

49 

6.83334937 

8.64:367107 

10.921.33313 

13.78384948 

17.37750403 

50 

7.10666335 

9.03263627 

11.46739979 

14.64196120 

18.42015429 

10 


n 

4% 

4V2% 

5% 

5V::7r 

6% 

51 

7.39095068 

9.43910490 

12.04076978 

15.34176907 

19.62636353 

52 

7.68658871 

9.86386Ui3 

12.64280826 

16.18556637 

20.69688534 

S3 

7.994-05256 

10.30773853 

13.27494868 

17.07577252 

21.93869846 

54 

8.31381 4.;35 

10.771.'58677 

13.93869611 

18.01494001 

23.25502037 

55 

8.64636692 

11.25630817 

14.63563092 

19.00576171 

24.650.32159 

56 

8.99222160 

11.76284204 

15.36741246 

20.05107860 

26.12934090 

57 

9.35191046 

12.29216993 

16.13578309 

21.1.5388793 

27.69710134 

58 

9.72598688 

12.84531758 

16.94257224 

22.317.35176 

29.35892742 

59 

10.11502635 

13.42335687 

17.78970085 

23.54480611 

31.12046307 

60 

10.51962741 

14.02740793 

18.67918589 

24.83977045 

32.98769085 

61 

10.94041250 

14.65864129 

19.61314519 

26.20596782 

34.96695230 

62 

11.37802900 

15.31828014 

20.59380245 

27.64728550 

37.06496944 

63 

11.83315016 

16.00760275 

21.62.349257 

29.16788620 

39.28886761 

64 

12.30647617 

16.72794487 

22.70466720 

30.77211994 

41.64619967 

65 

12.79873522 

17.48070239 

23.83990056 

32.46458654 

44.14497165 

66 

13.31068463 

18.26733400 

25.03189559 

34.25013880 

46.79366994 

67 

13.84311201 

19.08936403 

26.28349037 

36.13389643 

49.60129014 

68 

14.39683649 

19.94838541 

27.59766488 

38.12126074 

52.57736755 

69 

14.97270995 

20.84606276 

28.97754813 

40.21793008 

55.73200960 

70 

15.57161835 

21.78413558 

30.42642554 

42.42991623 

59.07693018 

71 

16.19448308 

22.76442168 

31.94774681 

44.76.356163 

62.62048699 

72 

16.84226241 

23.78882066 

33.54513415 

47.22555751 

66.37771515 

73 

1 7.51595290 

24.85931758 

35.22239086 

49.82296318 

70.36037806 

74 

18.21659102 

25.97798688 

36.98351040 

.52.56322615 

74.58200074 

75 

18.94525466 

27.14699629 

38.83268592 

55.45420359 

79.05692079 

76 

19.70306485 

28.36861112 

40.77432022 

58.50418479 

83.80033603 

77 

20.49118744 

29.64519862 

42.81303623 

61.72191495 

88.82835619 

78 

21.31083494 

30.97923256 

44.95368804 

65.11662027 

94.15805757 

79 

22.16326834 

32.37329802 

47.20137244 

68.69803439 

99.80754102 

80 

23.04979907 

33.83009643 

49.56144107 

72.47642628 

105.79699348 

81 

23.97179103 

35.35245077 

52.03951312 

76.46262973 

112.14375309 

82 

24.93066267 

36.94331106 

54.64148878 

80.66807436 

118.87237828 

83 

25.92788918 

38.60576006 

57.37356322 

85.10481845 

126.00472097 

84 

26.96500475 

40.34301926 

60.24224138 

89.78558347 

133.56500423 

.85 

28.04360494 

42.15845513 

63.25435344 

94.72379056 

141.57890449 

86 

29.16534914 

44.05558561 

66.41707112 

99.93359904 

150.07363875 

87 

30.33196311 

46.03808696 

69.73792467 

105.42994698 

159.07805708 

88 

31.54524163 

48.10980087 

73.22482091 

111.22859407 

168.62274050 

89 

32.80705129 

50.27474191 

76.88606195 

117.34616674 

178.74010493 

90 

34.11933334 

52..53710530 

80.73036505 

123.80020591 

189.46451123 

91 

35.48410668 

54.90127503 

84.76688330 

1.30.60921724 

200.82328190 

92 

86.90347094 

67.37183241 

89.00522747 

137.79272419 

212.88232482 

93 

38.37960978 

59.95356487 

93.45548884 

145.37132402 

226.65526431 

94 

39.91479417 

62.65147529 

98.12826328 

153.36674684 

239.19458017 

95 

41.51138594 

65.47079168 

103.03467645 

161.80191791 

253.54625498 

96 

43.17184138 

68.41697730 

108.18641027 

170.70102340 

268.75903027 

97 

44.89871503 

71.49574128 

113.59573078 

180.08957969 

284.88457209 

98 

46.69466363 

74.71304964 

119.27551732 

189.99450657 

301.97764642 

99 

48.56245018 

78.07513687 

125.23929319 

200.44420443 

320.09630520 

100 

50.50494818 

81.58851803 

131.50126785 

21 1 .46863567 

339.30208351 

11 


The  Present  Worth  of  One  Unit. 


V   — 

(1+i)" 

n 

1% 

1^4% 

11/2% 

1%% 

2% 

1 

.99009901 

.98765432 

.98522168 

.98280098 

.98029216 

2 

.98029605 

.97546106 

.97066175 

.96589777 

.96116878 

3 

.97059015 

.96341833 

.95031699 

.94928528 

.94232233 

4 

.9()0980.^4 

.95152428 

.94218423 

.93295851 

.92384543 

5 

.9514G5(i9 

.93977706 

.92826033 

.91691524 

.90573081 

6 

.94204524 

.92817488 

.91454219 

.90114254 

.88797138 

7 

.93271805 

.91671593 

.90102679 

.88564378 

.87056018 

8 

.92348322 

.90539845 

.88771112 

.87041157 

.85349037 

9 

.91433982 

.89422069 

.87459224 

.85544135 

.83675527 

10 

.90528694 

.88318093 

.86166723 

.84072860 

.82034830 

11 

.89632372 

.87227746 

.84893323 

.82626889 

.80426304 

12 

.88744923 

.86150860 

.83638742 

.81205788 

.78849318 

13 

.87866260 

.85087269 

.82402702 

.79809128 

.77303253 

14 

.86996297 

.84036809 

.81184928 

.78436490 

.75787502 

15 

.86134947 

.82999318 

.79985151 

.77087459 

.74301473 

16 

.85282126 

.81974635 

.78803104 

.75761631 

.72844581 

17 

.84437749 

.80962602 

.77638526 

.74458605 

.71416256 

18 

.83601731 

.79963064 

.76491159 

.73177990 

.70015937 

19 

.82773992 

.78975866 

.75360747 

.71919401 

.68643076 

20 

.81954447 

.78000855 

.74247042 

.70682458 

.67297133 

21 

.81143017 

.77037881 

.73149795 

.69466789 

.65977582 

22 

.80339621 

.76086796 

.72068763 

.68272028 

.64683904 

23 

.79544179 

.75147453 

.71003708 

.67097817 

.63415592 

24 

.78756613 

.74219707 

.69954392 

.65943800 

.62172149 

25 

.77976844 

.73303414 

.68920583 

.64809632 

.60953087 

26 

.77204796 

.72398434 

.67902052 

.63694970 

.59757928 

27 

.76440392 

.71504626 

.66898574 

.62599479 

.58586204 

28 

.75683557 

.70621853 

.65909925 

.61522829 

.57437455 

29 

.74934125 

.69749978 

.64935887 

.60464697 

.56311231 

30 

.74192292 

.68888867 

.63976243 

.59424764 

.55207089 

31 

.73457715 

.68038387 

.63030781 

.58402716 

.54124597 

32 

.72730411 

.67198407 

.62099292 

.57398247 

.53063330 

33 

.72010307 

.66368797 

.61181568 

.56411053 

.52022873 

34 

.71297734 

.65549429 

.60277407 

.55440839 

.51002817 

35 

.70591420 

.64740177 

.59386608 

.54487311 

.50002761 

36 

.69892495 

.63940916 

.58508974 

.53550183 

.49022315 

37 

.69200490 

.63151522 

.57644^309 

.52629172 

.48061093 

38 

.68515337 

.62371873 

.56792423 

.51724002 

.47118719 

39 

.67836967 

.61601850 

.55953126 

.50834400 

.46194822 

40 

.67165314 

.60841334 

.55126232 

.49960098 

.45289042 

41 

.66500311 

.60090206 

.54311559 

.49100834 

.44401021 

42 

.65841892 

.59348352 

.53508925 

.48256348 

.43530413 

43 

.65189992 

.58615656 

.52718153 

.47426386 

.42676875 

44 

.64544546 

.57892006 

.51939067 

.46610699 

.41840074 

45 

.63905492 

.57177290 

.51171494 

.45809040 

.41019680 

46 

.63272764 

.56471397 

.50415265 

.45021170 

.40215373 

47 

.62646301 

.55774219 

.49670212 

.4424()850 

.39426836 

48 

.62026041 

.55085649 

.48936170 

.43485848 

.38653761 

49 

.61411921 

.54405579 

.48212975 

.42737934 

.37895844 

50 

.60803882 

.53733905 

.47500470 

.42002883 

.37152788 

12 


n 

1% 

l'/4% 

^V2% 

1%% 

2% 

51 

.60201864 

.53070524 

.46798491 

.41280475 

.36424302 

52 

.59605806 

.52415332 

.46106887 

.40570492 

.35710100 

53 

.59015649 

.51768229 

.45425505 

.39872719 

.35009902 

54 

.584:31336 

.51129115 

.44754192 

.39186947 

.34323433 

55 

.57852808 

.50497892 

.44092800 

.38512970 

.33650425 

56 

.57280008 

.49874461 

.43441182 

.37850585 

.32990613 

57 

.56712879 

.49258727 

.42799194 

.37199592 

.32343740 

58 

.56151365 

.48650594 

.42166694 

.36559796 

.31709547 

59 

.5.5595411 

.48049970 

.41543541 

.35931003 

.31087791 

60 

.55044962 

.47456760 

.40929597 

.3.5313025 

.30478227 

61 

.54499962 

.46870874 

.40324726 

.34705676 

.29880614 

62 

.53960.358 

.46292222 

.39728794 

.34108773 

.29294720 

63 

.53426097 

.45720713 

.39141669 

.335221.35 

.28720314 

64 

.52897126 

.45156259 

.38563221 

.32945587 

.28157170 

65 

.52373392 

.44598775 

.37993321 

.32378956 

.27605070 

66 

.51854844 

.44048173 

.37431843 

.31822069 

.27063793 

67 

.51341429 

.43504370 

.36878663 

.31274761 

.26533130 

68 

.50833099 

.42967277 

.36333658 

.30736866 

.26012873 

69 

.50329801 

.42436817 

.35796708 

.30208222 

.25502817 

70 

.49831486 

.41912905 

.35267692 

.29688670 

.25002761 

71 

.493.38105 

.41395461 

.34746495 

.29178054 

.24512511 

72 

.48849609 

.40884406 

.34233000 

.28676221 

.24031874 

73 

.48365949 

.40379661 

.33727093 

.28183018 

.23560661 

74 

.47887078 

.39881147 

.33228663 

.27698298 

.23098687 

75 

.47412949 

.39388787 

.32737599 

.27221914 

.22645771 

76 

.46943514 

.38902506 

.32253793 

.26753724 

.22201737 

77 

.46478726 

.38422228 

.31777136 

.26293586 

.21766408 

78 

.46018541 

.37947879 

.31307523 

.25841362 

.21339616 

79 

.45562912 

.37479387 

.30844850 

.25396916 

.20921192 

80 

.45111794 

.37016679 

.30389015 

.24960114 

.20510973 

81 

.44665142 

.36559683 

.29939916 

.24530825 

.20108797 

82 

.44222913 

.36108329 

.29497454 

.24108919 

.19714507 

83 

.4378.5063 

.35662547 

.29061531 

.23694269 

.19327948 

84 

.43351547 

.35222268 

.28632050 

.23285761 

.18948968 

85 

.42922324 

.34787426 

.28208917 

.22886242 

.18577420 

86 

.42497350 

.34357951 

.27792036 

.22492621 

.18213157 

87 

.42076585 

.33933779 

.27381316 

.22105770 

.17856036 

88 

.41659985 

.33514843 

.26976666 

.21725572 

.17.505918 

89 

.41247510 

.33101080 

.26577996 

.21351914 

.17162665 

90 

.40839119 

.32692425 

.2618.5218 

.20984682 

.16826142 

91 

.40434771 

.32288814 

.25798245 

.20623766 

.16496217 

92 

.40034427 

.31890187 

.25416990 

.20269057 

.16172762 

93 

.39638046 

.31496481 

.25041369 

.19920450 

.15855649 

94 

.39245590 

.31107636 

.24671.300 

.19577837 

.15544754 

95 

.38857020 

.30723591 

.24306699 

.19241118 

.15239955 

96 

.38472297 

.30344287 

.23947487 

.18910190 

.14941132 

97 

38091383 

.29969666 

.23593583 

.18584953 

.14648169 

98 

.37714241 

.29599670 

.23244909 

.1826.5310 

.14360950 

99 

.37340832 

.29234242 

.22901389 

.17951165 

.14079363 

[00 

.36971121 

.28873326 

.22562944 

.17642422 

.13803297 

13 


n 

21/4% 

2V2% 

2%% 

3% 

3V2% 

1 
2 
3 
4 
5 

.97799511 

.97560976 

.97323601 

.97087379 

.96618357 

.95G47U4. 

.951814-U) 

.947188:33 

.94259591 

.93351070 

.9351'2732 

.92859941 

.92183779 

.91514166 

.90194271 

.9148i:3;55 

.90595064 

.89716573 

.88848705 

.87144223 

.894.71232 

.88385429 

.87315400 

.86260878 

.84197317 

Q 

.87502427 

.86229687 

.84978491 

.83748426 

.81350064 

7 

.85576946 

.84126524 

.82704128 

.81309151 

.78599096 

8 

.8;3()93835 

.82074657 

.80490635 

.78940923 

.75941156 

9 

.81852161 

.80072836 

.78336385 

.76641673 

.73373097 

10 

.80051013 

.78119840 

.76239791 

.74409391 

.70891881 

11 

.78289499 

.76214478 

.74199310 

.72242128 

.68494571 

12 

.76566748 

.74355589 

.72213440 

.70137988 

.66178330 

13 

.74881905 

.72542038 

.70280720 

.68095134 

.63940415 

14 

.73234137 

.70772720 

.68399728 

.66111781 

.61778179 

15 

.71622628 

.69046556 

.66569078 

.64186195 

.59689062 

16 

.70046580 

.67362493 

.64787424 

.62316694 

.57670591 

17 

.68505212 

.65719506 

.63053454 

.60501645 

.55720378 

18 

.66997763 

.64116591 

.61:365892 

.58739461 

.538:36114 

19 

.65523484 

.62552772 

.59723496 

.57028603 

.52015569 

20 

.64081647 

.61027094 

.58125057 

.55367575 

.50256588 

21 

.62671538 

.59538629 

.56569398 

.53754928 

.48557090 

22 

.61292457 

.58086467 

.55055375 

.52189250 

.46915063 

23 

.59943723 

.66669724 

.53581874 

.50669175 

.45328563 

24 

.58624668 

.55287535 

.52147809 

.49193374 

.4:3795713 

25 

.57334639 

.53939059 

.50752126 

.47760557 

.42314699 

26 

.56073000 

.52623472 

.49393796 

.46369478 

.40883767 

27 

.54839117 

.51339973 

.48071821 

.45018906 

.39501224 

28 

.53632388 

.50087778 

.46785227 

.43707675 

.38165434 

29 

.52452213 

.48866125 

.455:33068 

.42434636 

.36874815 

30 

.51298008 

.47674269 

.44314421 

.41198676 

.35627841 

31 

.50169201 

.46511481 

.43128391 

.39998714 

.34423035 

32 

.49065233 

.45377055 

.41974103 

.38833703 

.33258971 

33 

.47985558 

.44270298 

.40850708 

.37702625 

.32134271 

34 

.46929641 

.4:3190534 

.39757380 

.36604490 

.31047605 

35 

.45896959 

.42137107 

.38693314 

.35538340 

.29997686 

36 

.44887002 

.41109372 

.37657727 

.34503243 

.28983272 

37 

.4:3899268 

.40106705 

.36649856 

.33498294 

.2800:5161 

38 

.42933270 

.39128492 

.35668959 

.32522615 

.27056194 

39 

.41988530 

.38174139 

.:34714316 

.31575355 

.26141250 

40 

.41064575 

.37243062 

.33785222 

.30655684 

.25257247 

41 

.40160954 

.36334695 

.32880995 

.29762800 

.24403137 

42 

.39277216 

.35448483 

.32000968 

.28895922 

.2:3577910 

43 

.38412925 

.34583886 

.31144495 

.28054294 

.22780590 

44 

.37567653 

.33740376 

.:3031()944 

.27237178 

.22010231 

45 

.36740981 

.32917440 

.29499702 

.26443862 

.21265924 

46 

.35932500 

.32114576 

.28710172 

.25673653 

.20546787 

47 

.35101809 

.31:331294 

.27941773 

.24925876 

.19851968 

48 

.34:368518 

.:30567116 

.27193940 

.24199880 

.19180645 

49 

.:i36 12242 

.29821576 

.26466122 

.23495029 

.185:52024 

50 

.32872608 

.29094221 

.25757783 

.22810708 

.17905337 

14 


n 

2y4% 

2y2% 

2%  % 

3% 

^Vz7c 

51 

.32149250 

.28384606 

.25068402 

.22146318 

.172998-43 

52 

.31441810 

.27692298 

.24397471 

.21501280 

.16714824 

53 

.30749936 

.27016876 

.23744197 

.20875029 

.161-49589 

54 

.30073287 

.26357928 

.23109000 

.20267019 

.15603467 

55 

.29411528 

.25715052 

.22490511 

.1967671Z 

.15075814 

56 

.28764330 

.25087855 

.21888575 

.19103609 

.14566004 

57 

.28131374 

.24-475956 

.21302749 

.18547193 

.1-10734:34 

58 

.27512347 

.23878982 

.20732603 

.18006984 

.13597520 

59 

.26906940 

.23296568 

.20177716 

.17482508 

.13137701 

60 

.26314856 

.22728359 

.19637679 

.16973309 

.12693431 

61 

.25735801 

.22174009 

.19112097 

.16478941 

.12264184 

62 

.25169487 

.21633179 

.18600581 

.15998972 

.11849454 

63 

.24615635 

.21105541 

.18102755 

.15532982 

.11448747 

64 

.24073971 

.20590771 

.17618253 

.15080565 

.11061592 

65 

.23544226 

.20088557 

.17146718 

.14641325 

.10687528 

66 

.23026138 

.19598593 

.16687804 

.14214879 

.10326114 

67 

.22519450 

.19120578 

.16241172 

.13800853 

.09976922 

68 

.22023912 

.1865422.3 

.15806493 

.13398887 

.09639538 

69 

.21539278 

.18199241 

.15383448 

.13008628 

.09313563 

70 

.21065309 

.17755358 

.14971726 

.12629736 

.08998612 

71 

.20601769 

.17322300 

.14571023 

.12261880 

.08694311 

72 

.20148429 

.16899805 

.14181044 

.11904737 

.08400300 

73 

.19705065 

.16487615 

.13801503 

.11557998 

.08116232 

74 

19271458 

.16085477 

.13432119 

.11221357 

.07841770 

75 

.18847391 

.15693149 

.13072622 

.10894521 

.07576590 

76 

.18432657 

.15310389 

.12722747 

.10577205 

.07320376 

77 

.18027048 

.14936965 

.12382235 

.10269131 

.07072828 

78 

.17630365 

.14572649 

.12050837 

.09970030 

.06833650 

79 

.17242411 

.14217218 

.11728039 

.09679641 

.06602560 

80 

.16862993 

.13870457 

.11414412 

.09397710 

.06379285 

81 

.16491725 

.1.3532153 

.11108917 

.09123990 

.06163561 

82 

.16129022 

.13202101 

.10811598 

.08858243 

.05955131 

83 

.15774105 

.12880098 

.10522237 

.08600236 

.057537.50 

84 

.15426997 

.12565949 

.10240620 

.08349743 

.05559178 

85 

15087528 

.12259463 

.09966540 

.08106547 

.05371187 

86 

.14755528 

.11960452 

.09699795 

.07870434 

.05189553 

87 

.144:30835 

.11668733 

.09440190 

.07641198 

.05014060 

88 

.14113286 

.11384130 

.09187.533 

.07418639 

.04844503 

89 

.13802724 

.11106468 

.08941638 

.07202562 

.04680679 

90 

.13498997 

.10835579 

.08702324 

.06992779 

.04522395 

91 

.13201953 

.10571296 

.08469415 

.06789105 

.04369464 

92 

.12911445 

.1031.3460 

.08242740 

.06591364 

.04221704 

93 

.12627331 

.10061912 

.08022131 

.06399383 

.0!')78941 

94 

.12349468 

.09816.500 

.07807427 

.06212993 

03941006 

95 

.12077719 

.09577073 

.07598469 

.06032032 

.03807735 

96 

11811950 

.09343486 

.07395104 

.05856342 

.03678971 

97 

11.552029 

.09115.596 

.07197181 

.05685769 

.03554562 

98 

11297828 

.08893264 

.07004556 

.05.520614 

.03434359 

99 

.11049221 

.08676355 

.06817086 

.05359383 

.0:3318222 

100 

.10806084 

.08464737 

.06634634 

.05203284 

.03206011 

15 


n 

4% 

*V2% 

6% 

5V2% 

6% 

1 

.96153846 

.95693780 

.95238095 

.94786730 

.94339623 

2 

.9245.'j621 

.91572995 

.90702948 

.89845242 

.88999644 

3 

.88899636 

.87629660 

.86383760 

.85161366 

.83961928 

4 

.85480419 

.83856134 

.82270427 

.80721674 

.79209366 

5 

.82192711 

.80245105 

.78352617 

.765134:35 

.74725817 

6 

.79031453 

.76789574 

.74621540 

.72524583 

.70496054 

7 

.75991781 

.73482846 

.71068133 

.68743681 

.66505711 

8 

.73069021 

.70318513 

.67683936 

.65159887 

.62741237 

9 

.702.')8674 

.67290443 

.64460892 

.61762926 

.59189846 

10 

.67556417 

.64392768 

.61391325 

.58543058 

.55839478 

11 

.64958093 

.61619874 

.58467929 

.55491051 

.52678753 

12 

.62459705 

.58966386 

.55683742 

.52598152 

.49696936 

13 

.60057409 

.56427164 

.53032135 

.49856068 

.46883902 

14 

.57747508 

.53997286 

.50506795 

.47256937 

.44230096 

15 

.55526450 

.51672044 

.48101710 

.44793305 

.41726506 

16 

.53390818 

.49446932 

.45811152 

.42458019 

.39364628 

17 

.51337323 

.47317639 

.43629669 

.40244653 

.37136442 

18 

.49362812 

.45280037 

.41552065 

.38146590 

.35034379 

19 

.47464242 

.43330179 

.39573396 

.36157906 

.33051301 

20 

.45638695 

.41464286 

.37688948 

.34272896 

.31180473 

21 

.43883360 

.39678743 

.35894236 

.32486158 

.29415540 

22 

42195539 

.37970089 

.34184987 

.30792567 

.27750510 

23 

.40572638 

.36335013 

.32557131 

.29187267 

.26179726 

24 

39012147 

.34770347 

.31006791 

.27665656 

.24697855 

25 

.37511680 

.33273060 

.29530277 

.26223370 

.23299863 

26 

.36068923 

.31840248 

.28124073 

.24856275 

.21981003 

27 

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.30469137 

.26784832 

.23560450 

.20736795 

28 

.33347747 

.29157069 

.25509364 

.22332181 

.19563014 

29 

.32065141 

.27901502 

.24294632 

.21167944 

.18455674 

30 

.30831867 

.26700002 

.23137745 

.20064402 

.17411013 

31 

.29646026 

.25550241 

.22035947 

.19018390 

.16425484 

32 

.28505794 

.24449991 

.20986617 

.18026910 

.15495740 

33 

.27409417 

.23397121 

.19987254 

.17087119 

.14618622 

34 

.26355209 

.22389589 

.19035480 

.16196321 

.13791153 

35 

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.21425445 

.18129029 

.15351963 

.13010522 

36 

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.20502817 

.17265741 

.14551624 

.12274077 

37 

23429685 

.19619921 

.16443563 

.13793008 

.11579318 

38 

.2252854;3 

.18775044 

.15660536 

.13073941 

.10923885 

39 

.21662061 

.17966549 

.14914797 

.12392362 

.103055.52 

40 

.20828904 

.17192870 

.14204.568 

.11746314 

.09722219 

41 

.20027793 

.164,52507 

.13528160 

.11133947 

.09171905 

42 

.19257499 

.15714026 

.12883962 

.10553504 

.08652740 

43 

.18516820 

.15066054 

.12270440 

.10003322 

.08162962 

44 

17804635 

.14417276 

.11686133 

.09481822 

.07700908 

45 

.17119841 

.13796437 

.11129651 

.08987509 

.07265007 

46 

.16461386 

.13202332 

.10599668 

.08518965 

.06853781 

47 

.15828256 

.12633810 

.10094921 

.08074849 

.06465831 

48 

.15219476 

.12089771 

.09614211 

.07653885 

.06099840 

49 

.14634112 

.11569158 

.09156391 

.07254867 

.05754.566 

50 

. 14071262 

.11070965 

.08720373 

.06876652 

.05428836 

16 


n 

4% 

*V2% 

5% 

5V2% 

6% 

51 

.13530059 

.10594225 

.08305117 

.06518153 

.05121544 

52 

.13009072 

.10138014 

.07909635 

.06178344 

.04831645 

53 

.12509300 

.09701449 

.07532986 

.05856250 

.04558156 

54 

.12028273 

.09283683 

.07174272 

.05550948 

.0-4300147 

55 

115G5551 

.08883907 

.06832640 

.05261562 

.04056742 

56 

.11120722 

.08501347 

.06507276 

.04987263 

.03827115 

57 

.10093002 

.0813.5260 

.06197406 

.0472726.3 

.03610486 

58 

.10281733 

.07784938 

.05902291 

.04-480818 

.03406119 

59 

.0988()282 

.07449701 

.05621230 

.04247221 

.03213320 

60 

.09506040 

.07128901 

.05353552 

.04025802 

.03031434 

61 

.0911.04.23 

.06821915 

.05098621 

.03815926 

.02859843 

62 

.08788868 

.06528M* 

.04855830 

.03616992 

.02697965 

63 

.08450835 

.06247032 

.04624600 

.03428428 

.02545250 

64 

.08125803 

.05978021 

.04404381 

.03249695 

.02401179 

65 

.07813272 

.05720594 

.04194648 

.03080279 

.02265264 

66 

.07512762 

.05474253 

.03994903 

.02919696 

.02137041 

67 

.07223809 

.05238519 

.03804670 

.02767485 

.02016077 

68 

.06945970 

.05012937 

.03623495 

.02623208 

.019019.59 

69 

.06678818 

.04797069 

.03450948 

.02486453 

.01794301 

70 

.06421940 

.04590497 

.03286617 

.02356828 

.01692737 

71 

.06174942 

.04392820 

.03130111 

.02333960 

.01596921 

72 

.05937445 

.04203655 

.02981058 

.02117498 

.01506530 

73 

.05709081 

.04022637 

.02839103 

.02007107 

.01421254 

74 

.05489501 

.03849413 

.02703908 

.01902471 

.01,340806 

75 

.05278367 

.03683649 

.02575150 

.01803290 

.01264911 

76 

.05075353 

.03525023 

.02452524 

.01709279 

.0119.3313 

77 

.04880147 

.03373228 

.02335737 

.01620170 

.01125767 

78 

.04692449 

.03227969 

.02224.512 

.01535706 

.01062044 

79 

.04511970 

.03088965 

.02118582 

.014.55646 

.01001928 

80 

.04838433 

.02955948 

.02017698 

.01379759 

.00945215 

81 

.04171570 

.02828658 

.01921617 

.01307828 

.00891713 

82 

.04011125 

.02706850 

.01830111 

.01239648 

.00841238 

83 

.03856851 

.02590287 

.01742963 

.01175022 

.00793621 

84 

.03708510 

.02478744 

.016.59965 

.01113765 

.00748699 

85 

.03565875 

.02372003 

.01580919 

.01055701 

.00706320 

86 

.03428726 

.02269860 

.01595637 

.01000664 

.00666340 

87 

.03296852 

.02172115 

.01433940 

.00948497 

.00628622 

88 

.03170050 

.02078579 

.01365657 

.00899049 

.00593039 

89 

.03048125 

.01989070 

.01.300626 

.00852180 

.00559472 

90 

.02930890 

.01903417 

.01238691 

.ooscrycs 

.00527803 

91 

.02818163 

.01821451 

.01179706 

.00765643 

.00497928 

92 

.02709772 

.01743016 

.01123530 

.00725728 

.00469743 

93 

.02605550 

.01667958 

.01070028 

.00687894 

.00443154 

94 

.02505337 

.01.596132 

.01019074 

.00652032 

.00418070 

95 

.02408978 

.01527399 

.00970547 

.00618040 

.00394405 

96 

.02316325 

.01461626 

.00924331 

.00585820 

.00372081 

97 

.02227235 

.01398685 

.00880315 

.00555280 

.00351019 

98 

.02141572 

.01338454 

.00838395 

.0052():331 

.00331150 

99 

.02059204 

.01280817 

.00798471 

.00498892 

.00312406 

100 

.01980004 

.01225663 

.00760449 

.00472883 

.00294723 

17 


The  Amount  of  Pekiodic  Payments  of  One  Unit  Each. 

s'^— 1 


n 

1% 

o  — 
n 

iy4% 

i 

1%% 

1%% 

2% 

1 

1.00 

1.00 

1.00 

1.00 

1.00 

2 

2.01 

2.0125 

2.015 

2.0175 

2.02 

3 

3.0301 

3.03765625 

3.045225 

3.06280626 

3.0604 

4 

4.060401 

4.07562695 

4.09090338 

4.10623036 

4.121608 

5 

5.10100501 

5.12657229 

5.16226693 

6.17808939 

6.20404016 

6 

6.15201506 

6.19065444 

6.22955093 

6.26870596 

6.30812096 

7 

7.21353521 

7.26803762 

7.32299419 

7.37840831 

7.413428338 

8 

8.28567056 

8.35888809 

8.43283911 

8.50753046 

8.68296905 

9 

9.36852727 

9.46337420 

9.55933169 

9.66641224 

9.76462843 

10 

10.46221524 

10.58166637 

10.70272167 

10.82639946 

10.94972100 

11 

11.56684367 

11.71393720 

11.86326249 

12.01484394 

12.16871642 

12 

12.68250301 

12.86036142 

13.04121143 

13.22510371 

13.41208973 

13 

13.80932804 

14.02111594 

14.23682960 

14.45654303 

14.68033152 

14 

14.94742132 

15.19637989 

15.45038205 

15.70953253 

16.97393815 

15 

16.09689554 

16.38633463 

16.68213778 

16.98444935 

17.29341692 

16 

17.25786449 

17.59116382 

17.93236984 

18.28167721 

18.63928625 

17 

18.43044314 

18.81105336 

19.20135539 

19.60160656 

20.01207096 

18 

19.61474757 

20.04611953 

20.48937572 

20.94463468 

21.41231238 

19 

20.81089504 

21.29676893 

21.79671636 

22.31116579 

22.84055863 

20 

22.09100399 

22.56297854 

23.12366710 

23.70161119 

24.29736980 

21 

23.23919403 

23.84.501577 

24.47052211 

25.11638938 

26.78331719 

22 

24.47158598 

25.14307847 

25.83757994 

26.55592620 

27.29898364 

23 

25.71630183 

26.45736695 

27.22514364 

28.02065490 

28.84496321 

24 

26.97346485 

27.78808403 

28.63352088 

29.61101637 

30.42186247 

25 

28.24319950 

29.13543508 

30.06302361 

31.02736915 

32.03029972 

26 

29.52563150 

30.49962802 

31.51396897 

32.67043969 

33.67090672 

27 

30.82087781 

31.88087337 

32.98667850 

34.14042238 

35.34432383 

28 

32.12909669 

33.27938428 

34.48147868 

36.73787977 

37.06121031 

29 

33.45038766 

34.69537659 

35.99870086 

37.36329266 

38.79223461 

30 

34.78489153 

36.12906880 

37.63868137 

39.01716029 

40.56897921 

31 

36.13274045 

87.58068216 

39.10176159 

40.69995042 

42.37944079 

32 

37.49406785 

39.05044068 

40.68828801 

42.41219955 

44.22702961 

33 

38.861f)0()853 

40.53857120 

42.29861233 

44.15441306 

46.11157020 

34 

40.25769863 

42.04530334 

43.93309152 

45.92711527 

48.03380160 

35 

41.66027560 

43.57086963 

45.59208789 

47.73083979 

49.99447763 

36 

43.07687836 

45.11550550 

47.27596921 

49.66612949 

51.99436719 

37 

44.50764714 

46.67944932 

48.98510875 

61.43353675 

54.03426453 

38 

45.95272361 

48.26294243 

50.71988538 

63.33362365 

66.11493962 

39 

47.41225085 

49.86622921 

52.48068366 

56.26696206 

68.23723841 

40 

48.88637336 

51.48955708 

54.26789392 

67.23413390 

60.40198318 

41 

50.37523709 

63.13317654 

56.08191232 

59.23573124 

62.61002284 

42 

51.87898946 

54.79734125 

57.92314100 

61.27236654 

64.86222330 

43 

53.39777935 

66.48230802 

59.79198812 

63.34462278 

67.15946777 

44 

54.93175715 

58.18833687 

61.68886794 

65.45316368 

69.50266712 

45 

56.48107472 

59.91569108 

63.61420096 

67.59858386 

71.89271027 

46 

58.01.588547 

61.66463722 

65.66841397 

69.78955908 

74.33056447 

47 

59.(i26344;32 

63.43544518 

67.55194018 

72.00273637 

76.81717576 

48 

61.22260777 

65.22838824 

69.66521928 

74.26278426 

79.35351927 

49 

62.831.S3385 

67.04;374310 

71.60869757 

76.66238298 

81.94058966 

50 

64.46318218 

68.88178989 

73.68282804 

78.90222468 

84.57940145 

18 


n 

1% 

iy4% 

iy2% 

iy4% 

2% 

51 

66.10781401 

70.74281227 

75.78807046 

81.28301361 

87.27098948 

52 

67.76889215 

72.62709742 

77.92489152 

83.705.4()635 

90.01640927 

53 

69.44658107 

74.53493613 

80.09376489 

86.17031201 

92.81673745 

54 

71.14104688 

76.46662284 

82.29517136 

88.67829247 

95.67307221 

55 

72.85245735 

78.42245562 

84.52959893 

91.23016259 

98.59653365 

56 

74.58098193 

80.40273633 

86.79754292 

93.8266904:3 

101.558264:32 

57 

76.32679175 

82.40777053 

89.09950606 

96.4()865752 

104.58942960 

58 

78.09005966 

84.43786766 

91.43599865 

99.15685902 

107.68121819 

59 

79.87096026 

86.49334101 

93.80753863 

101.89210405 

110.83484256 

60 

81.66966986 

88.57450777 

96.21465171 

104.67521586 

114.0515:3942 

61 

83.48636656 

90.68168911 

98.65787149 

107.5070:3215 

117.;3:3257021 

62 

85.32123023 

92.81521022 

101.13773956 

110.38840522 

120.67922161 

63 

87.17444253 

94.97540035 

103.65480565 

113.32020231 

124.09280604 

64 

89.04618695 

97.16259285 

106.20962774 

116.:30330585 

127.57466216 

65 

90.93664882 

99.37712526 

108.80277215 

119.33861370 

131.12615541 

66 

92.84601531 

101.61933934 

111.43481374 

122.42703944 

134.74867851 

67 

94.77447546 

103.88958108 

114.10633594 

125.56951263 

138.44:365208 

68 

96.72222022 

106.18820084 

116.81793098 

128.76697910 

142.21252513 

69 

98.68944242 

108.51555355 

119.57019995 

132.02040124 

146.05677563 

70 

100.67633684 

110.87199776 

122.36375295 

135.33075826 

149.97791114 

71 

102.68310021 

113.25789773 

125.19920924 

138.69904653 

153.97746936 

72 

104.70993121 

115.67362145 

128.07719738 

142.12627984 

158.05701875 

73 

106.75703053 

118.11954172 

130.99835534 

145.61348974 

162.21815913 

74 

108.82460083 

120.59603599 

133.96333067 

149.16172581 

166.46252231 

75 

110.91284684 

123.10348644 

136.97278063 

152.77205601 

170.79177276 

76 

113.02197531 

125.64228002 

140.02737234 

156.44556699 

175.20760822 

77 

115.15219506 

128.21280852 

14:5.12778293 

160.18336441 

179.711760:38 

78 

117.30371701 

130.81546862 

146.27469967 

163.98657:329 

184.30599559 

79 

119.47675418 

133.45066199 

149.46882017 

167.85633832 

188.99211550 

80 

121.67152172 

136.11879526 

152.71085247 

171.79382424 

193.77195780 

81 

123.88823694 

138.82028020 

156.00151526 

175.80021617 

198.64739697 

82 

126.12711931 

141.55553371 

159.34153799 

179.87671995 

203.62034490 

83 

128.38839050 

144.32497788 

162.73166106 

184.02456255 

208.69275181 

84 

130.67227441 

147.12904010 

166.17263597 

188.24499239 

213.86660684 

85 

132.97899715 

149.96815310 

169.66522552 

192.53927976 

219.14393898 

86 

135.30878713 

152.84275501 

173.21020390 

196.90871716 

224.52681776 

87 

137.66187500 

155.75328945 

176.80835696 

201.35461971 

230.017:35411 

88 

140.03849375 

158.70020556 

180.46048231 

205.87832555 

235.61770119 

89 

142.43887868 

161.68395813 

184.16738955 

210.48119625 

241.;3:3005521 

90 

144.86326745 

164.70500761 

187.92990039 

215.16461718 

247.15665632 

91 

147.31190014 

167.76382021 

191.74884889 

219.92999798 

253.09978945 

92 

149.78501914 

170.86086796 

195.62508162 

224.77877295 

259.16178523 

93 

152.28286933 

173.99662881 

199.55945784 

229.71240146 

265.34502094 

94 

154.80569802 

177.17158667 

203.55284971 

234.73236850 

271.65192135 

95 

157.55375501 

180.38623150 

207.60614246 

239.84018495 

278.08495978 

96 

159.92729255 

183.64105941 

211.72023459 

245.03738819 

284.64665898 

97 

162.52656548 

186.93657265 

215.89603811 

250.32554250 

291.:3:3959216 

98 

165.15183111 

190.27327980 

220.13447868 

255.70623947 

298.16638400 

99 

167.80334945 

193.65169580 

224.4:3649586 

261.18109866 

305.12971168 

100 

170.48138294 

197.07234200 

228.80304330 

266.75176789 

312.23200591 

19 


n 

2y4% 

2y2% 

2%% 

3% 

3y2% 

1 

1.00 

1.00 

1.00 

1.00 

1.00 

2 

2.022.5 

2.025 

2.0275 

2.03 

2.035 

3 

3.0fiS00f)2.5 

3.075625 

3.08325625 

3.0909 

3.106226 

4 

4..187036.39 

4.1.5251563 

4.16804580 

4.183627 

4.2149428S 

5 

5.23011971 

6.25632852 

5.28266706 

6.30913581 

6.3024ti5^>S 

6 

fi.3.1.779740 

6.38773673 

6.42794040 

6.4<i840988 

6.55015218 

7 

7.49062284 

7.5474.3016 

7.60470876 

7.66246218 

7.77940751 

8 

8.6.591(5186 

8.7.3611590 

8.81383825 

8.89233605 

9.05168677 

9 

9.85399300 

9.95451880 

10.05621880 

10.15910613 

10.36819581 

10 

11.07570784 

11.20338177 

11.33276482 

11.46387931 

11.73139316 

11 

12.32491127 

12.48346631 

12.64441585 

12.80779569 

1.3.14J  99192 

12 

13.60222177 

13.79555297 

13.99213729 

14.19202956 

14.60196164 

13 

14.90827176 

1,5.14041179 

15.37692107 

15.61779045 

16.11303030 

.14 

16.24370788 

16.51895284 

16.79978639 

17.086.32416 

17.676986.36 

15 

17.60919131 

17.93192666 

18.26178052 

18.59891389 

19.29668088 

16 

19.00.539811 

19.38022488 

19.76.397948 

20.15688130 

20.97102971 

17 

20.43301957 

20.86473045 

21. .30748892 

21.76158774 

22.70.501575 

18 

21.89276251 

22.38634871 

22.89344487 

23.41443537 

24.49969130 

19 

23.38534966 

23.94600743 

24.52301460 

25.11686844 

26.357180.50 

20 

24.91152003 

25.54465761 

26.19739750 

26.87037449 

28.27968181 

21 

26.47202923 

27.18327405 

27.91782593 

28.67648572 

30.26047068 

22 

28.06764989 

28.86285590 

29.68556616 

30.53678030 

32..32890215 

23 

29.69917201 

30.58442730 

31.50191921 

32.45288370 

34.46041374 

24 

31.36740338 

32.34903798 

33.36822199 

34.42647022 

36.666.52821 

25 

33.07316996 

34.15776393 

35.28584810 

36.46926432 

38.94986669 

.26 

34.81731628 

36.01170803 

37.25620892 

38.65304225 

41.31310168 

27 

36.60070590 

37.91200073 

39.28075467 

40.70963352 

43.75906024 

28 

38.42422178 

39.85980075 

41.36097542 

42.93092252 

46.29062734 

29 

40.28876677 

41.85629577 

43.49840224 

45.21885020 

48.91079930 

30 

42.19526402 

43.90270316 

45.69460831 

47.67541571 

61.62267728 

31 

44.14465746 

46.00027074 

47.95121003 

60.00267818 

64.42947098 

32 

46.13791226 

48.15027752 

50.26986831 

52.50275852 

67.33450247 

33 

48.17601528 

60.35403445 

52.65228969 

65.07784128 

60.34121005 

34 

50.25997563 

52.61288531 

65.10022765 

67.73017652 

63.45316240 

35 

52.39082508 

54.92820745 

57.61548391 

60.46208181 

66.47401274 

36 

54.56961864 

57.30141263 

60.19990972 

63.27594427 

70.00760318 

37 

56.79743506 

59.73394794 

62.85540724 

66.17422259 

73.45786930 

38 

59.07537735 

62.22729665 

65.58393094 

69.15944927 

77.02889472 

39 

61.40457334 

64.78297907 

68.38748904 

72.23422376 

80.72490604 

40 

63.78617624 

67.40255354 

71.26814499 

75.40125973 

84.55027776 

41 

66.22136521 

70.08761737 

74.22801897 

78.66329753 

88.60953748 

42 

68.71134592 

72.83980780 

77.26928950 

82.02319646 

92.60737129 

43 

71.25735121 

75.66080300 

80.39419496 

85.48389234 

96.84862928 

44 

7.3.86064161 

78.552.32308 

83.60503534 

89.04840911 

101.23833131 

45 

76.52250606 

81.51613116 

86.90417379 

92.71986138 

105.78167290 

46 

79.24426243 

84.55403443 

90.294.03857 

96.50145723 

110.48403145 

47 

82.027258.34 

87.66788530 

93.77712463 

100.39()50095 

115.35097255 

48 

84.87287165 

90.85958242 

97.35599556 

104.40839598 

120.38825659 

49 

87.78251126 

94.13107199 

101.03328545 

108.54064785 

125.60184557 

50 

90.76751776 

97.48434879 

104.81170079 

112.79686729 

130.99791016 

20 


n 

2V4% 

2y2% 

2%% 

3% 

31/2% 

51 

93.799G6416 

100.92145751 

108.69402256 

117.18077331 

136.58283702 

52 

96.91015()61 

104.44449394 

112.68310818 

121.69619651 

142.36323631 

53 

100.090():35i;3 

108.05560629 

116.78189365 

126.34708240 

148.34594958 

54 

10;3.342()7t.42 

111.75799645 

120.99339573 

131.13749488 

154.-53805782 

55 

10G,6G7884G0 

115.55092136 

125.32071411 

136.07161972 

160.94688984 

56 

110.0()791200 

119.43969439 

129.76703375 

141.15376831 

167.58003099 

57 

lVi.5Ul'U)00 

123.42568765 

134.33562717 

146.38838136 

174.44-533207 

58 

117.09918992 

127.51132892 

139.02985692 

151.78003280 

18I.-5-'>091869 

59 

120.73;392169 

131.69911214 

143.85317799 

157.3334:3379 

188.90520085 

60 

124.45043493 

135.99158995 

148.80914038 

163.05343680 

196.51688288 

61 

128.25056972 

140.39137969 

153.90139174 

168.94503991 

204.39497378 

62 

132.13620753 

144.90116419 

159.13368002 

175.01339110 

212.-5487978G 

63 

136.10927220 

149.52369329 

164.50985622 

181.26379284 

220.98800579 

64 

140.17173083 

154.26178563 

170.03387726 

187.70170662 

229.72258599 

65 

144.32559477 

159.11833027 

175.70980889 

194.33275782 

238.76287650 

66 

148.57292605 

164.09628852 

181.54182863 

201.16274055 

248.11957718 

67 

152.91581137 

169.19869574 

187.53422892 

208.19762277 

257.80376238 

68 

157.35641712 

174.42866313 

193.69142022 

215.44355145 

267.82689406 

69 

161.89693651 

179.78937971 

200.01793427 

222.90685800 

278.20083535 

70 

166.53961758 

185.28411421 

206.51842746 

230.59406374 

288.93786459 

71 

171.28675898 

190.91621706 

213.19768422 

238.51188565 

300.05068985 

72 

176.14071105 

196.68912249 

220.06062054 

246.66724222 

311.55246400 

73 

181.10387705 

202.60635056 

227.11228760 

255.06725949 

323.45680023 

74 

186.17871429 

208.67150932 

234.35787551 

263.71927727 

3-35.77778824 

75 

191.36773538 

214.88829706 

241.80271709 

272.63085559 

348.53001083 

76 

196.67350940 

221.26050449 

249.45229181 

281.80978126 

361.72856121 

77 

202.09866337 

227.79201710 

257.31222983 

291.26407469 

375.38906085 

78 

207.64588329 

234.48681752 

265.38831615 

301.00199693 

389.52767798 

79 

213.31791567 

241.34898795 

273.68649485 

311.03205684 

404.16114671 

80 

219.11756877 

248.38271265 

282.21287345 

321.36301855 

419.30678685 

81 

225.04771497 

255.59228047 

290.97372747 

332.00390910 

434.98252439 

82 

231.11128763 

262.98208748 

299.97550499 

342.96402638 

451.20691274 

83 

237.31129160 

270.55663967 

309.22483137 

354.25294717 

467.99915469 

84 

243.65079567 

278.32055566 

318.72851423 

365.88053558 

485.37912510 

85 

250.13293857 

286.27856955 

328.49354837 

377.85695165 

503.36739448 

86 

256.76092969 

294.43553379 

338.52712095 

390.19266020 

521.98525329 

87 

263.53805060 

302.79642214 

348.83661678 

402.89844001 

541.25473715 

88 

270.46765674 

311.36633269 

359.42962374 

415.98539321 

561.19865295 

89 

277.55317902 

320.15041900 

370.31393839 

429.46495500 

581.84060581 

90 

284.79812555 

329.15425328 

381.49757170 

443.34890365 

603.20502701 

91 

292.20608337 

338.38310961 

392.98875492 

457.64937076 

625.31720296 

92 

299.78072025 

347.84268735 

404.79594568 

472.37885189 

648.20330506 

93 

307.52578644 

357.53875453 

416.92783418 

487.55021745 

671.89042074 

94 

315.44511665 

367.47722340 

429.39334962 

503.17672397 

696.40658546 

95 

323.54263177 

377.66415398 

442.20166674 

519.27202569 

721.78081595 

96 

331.82234099 

388.10575783 

455.36221257 

535.85018646 

748.04.314451 

97 

340.28834366 

398.80840178 

468.884.67342 

552.92569205 

775.22465457 

98 

348.94483139 

409.77861182 

482.77900194 

570.51:546282 

803.35751748 

99 

357.79609010 

421.02307712 

497.05542449 

588.62886670 

832.47503059 

100 

366.84650213 

432.54865404 

511.72444867 

607.28773270 

862.61165066 

21 


n 

4% 

*V2% 

5% 

51/2% 

6% 

1 

1.00 

1.00 

1.00 

1.00 

1.00 

2  . 

2.04 

2.045 

2.0o 

2.055 

2.06 

3 

3.121G 

3.137025 

3.1525 

3.168025 

3.1836 

4 

4.246164 

4.27819113 

4.310125 

4.34226638 

4.374616 

5 

6.41632256 

6.47070973 

6..52563125 

5..58109103 

5.63709296 

6 

6.63297546 

6.71689166 

6.80191281 

6.88805103 

6.97531854 

7 

7.89829418 

8.01915179 

8.14200845 

8.26689384 

8.39383765 

8 

9.21422626 

9.38001362 

9.54910888 

9.72157300 

9.89746791 

9 

10.58279531 

10.80211423 

11.02656432 

11.2.562,5951 

11.49131598 

10 

12.00610712 

12.28820937 

12.577892.54 

12.87535379 

13.18079494 

11 

13.48635141 

13.84117879 

14.20678716 

14.58349825 

14.97164264 

12 

15.02580546 

15.46403184 

15.91712652 

16.38559065 

16.86994120 

13 

16.62683768 

17.1.5991.327 

17.71298284 

18.29679814 

18.88213767 

14 

18.29191119 

18.93210937 

19.59863199 

20.29257203 

21.01506593 

15 

20.02358764 

20.78405429 

21.57856359 

22.40866250 

23.27596988 

16 

21.82453114 

22.71933673 

23.65749177 

24.64113999 

25.67252808 

17 

23.69751239 

24.74170689 

25.84036636 

26.99640269 

28.21287976 

18 

25.64541288 

26.85508370 

28.13238467 

29.48120483 

30.9056,5255 

19 

27.67122940 

29.06356246 

30.53900391 

32.10267110 

33.75999170 

20 

29.77807858 

31.37142277 

33.06595410 

34.86831801 

36.78559120 

21 

31.96920172 

33.78313680 

35.71925181 

37.78607,550 

39.99272668 

22 

34.24796979 

36.30.337795 

38.50521440 

40.86430965 

4:3.39229028 

23 

36.617888.58 

38.93702996 

41.43047512 

44.11184669 

46.99582769 

24 

39.08260412 

41.68919631 

44.50199887 

47.53799825 

50.81557735 

25 

41,64590829 

44.56521015 

47.72709882 

51.15258816 

54.86451200 

28 

44.31174462 

47.57064460 

51.11345376 

54.96598051 

59.15638272 

27 

47.08421440 

50.71132361 

54.66912645 

58.9891094:3 

63.70576568 

28 

49.96758298 

53.99.333319 

58.40258277 

63.23351045 

68.52811162 

29 

52.96628630 

57.42303316 

62.32271191 

67.711,3,53.53 

73.63979832 

30 

66.08493775 

61.00706966 

66.43884750 

72.43547797 

79.05818622 

31 

59.32833526 

64.75238779 

70.76078988 

77.41942926 

84.80167739 

32 

62.70146867 

68.66624524 

75.29882937 

82.67749787 

90.88977803 

33 

66.20952742 

72.75622628 

80.06377084 

88.22476025 

97.34316471 

34 

69.857.90851 

77.03025646 

85.06695938 

94.07712206 

104.18375459 

35 

73.65222486 

81.49661800 

90.32030735 

100.251.36378 

111.43477987 

36 

77.59831385 

86.16396-581 

95.83632272 

106.76518879 

119.12086666 

37 

81.70224640 

91.04134427 

101.62813886 

113.63727417 

127.26811866 

38 

85.97033626 

96.13820476 

107.70954580 

120.88732425 

135.90420578 

39 

90.40914971 

101.46442.398 

114.09502309 

128.53612708 

145.05845813 

40 

95.02551570 

107.03032306 

120.79977424 

136.60561407 

154.76196562 

41 

99.82653633 

112.84668760 

127.83976295 

145.11892285 

165.04768356 

42 

104.81959778 

118.92478854 

135.23175110 

154.10046360 

175.95054457 

43 

110.012.38169 

125.27640402 

142.99.3.33866 

16.3.57,598910 

187.50757724 

44 

11.5.41287696 

131.91384220 

151.1^300559 

173.572668.50 

199.75803188 

45 

121.02939204 

138.84996510 

159.70015587 

184.11916,527 

212.74351:379 

46 

126.87056772 

146.09821353 

168.68516366 

195.24571936 

226.50812462 

47 

132.94539043 

153.67263314 

178.11942185 

206.98423392 

241.09861210 

48 

139.26320604 

161.58790163 

188.02539294 

219.36836679 

256.56452882 

49 

145.8337;}  1.29 

169.85935720 

198.42666259 

232.43362696 

272.95840055 

50 

152.66708366 

178.60302828 

209.34799572 

246.21747645 

290.33590458 

22 


n 

4% 

*V2% 

5% 

5%% 

6% 

51 

159.77:370700 

187.53560455 

220.81539550 

260.75943765 

308.75605886 

52 

167.1G1.71768 

196.97476940 

2:32.85616528 

276.10120672 

328.28142239 

53 

174.851306.39 

200.83803408 

245.49897354 

292.28677.309 

348.97830773 

54 

182.84.5358fi5 

217.14637261 

258.77392222 

309.30254501 

370.91700620 

55 

191.15917299 

227.91795938 

272.71261833 

327.37748562 

394.17202657 

56 

199.80553991 

239.17426756 

287.34824924 

346.38.324733 

418.82234810 

57 

208.7977fil51 

250.93710959 

302.71566171 

360.4.343259:} 

444.95168905 

58 

218.149G7197 

263.22927953 

318.85144479 

387.58821:380 

472.04879040 

59 

227.87505885 

270.07459711 

3:35.7940170:} 

409.90.550562 

502.00771782 

60 

237.9900-8520 

289.49795398 

353.58371788 

433.45037173 

533.12818089 

61 

248.51031261 

303.52536190 

372.26290378 

458.29014217 

566.11587174 

62 

259.45072511 

318.18400319 

391.87604897 

484.49609999 

601.08282405 

63 

270.82875412 

333.50228333 

412.46985141 

512.14338549 

6.38.14779:349 

64 

282.00190428 

349.50988608 

434.09334398 

541.31127109 

677.43000110 

65 

294.90838045 

366.23783096 

456.79801118 

572.08339104 

719.08286070 

66 

307.70711567 

383.71853335 

480.63791174 

604.54797818 

763.22783241 

67 

321.07780030 

401.98580735 

.505.66980733 

0.38.79811098 

810.021502:50 

68 

334.92091231 

421.07523138 

.531.95.329770 

074.93201341 

859.62279250 

69 

349.31774880 

441.02301079 

559.55090258 

713.05327415 

912.20016005 

70 

364.29045876 

401.80907955 

588.52851071 

753.27120423 

967.93216965 

71 

379.80207711 

483.05381513 

018.9549.3025 

795.70112040 

1027.00809983 

72 

396.05050019 

506.41823681 

050.90208300 

840.40408208 

1089.02858582 

73 

412.89882200 

530.20705747 

084.44781721 

887.69023960 

1150.000.30097 

74 

430.41477550 

555.06037505 

719.08020807 

937.51320278 

1220.36667902 

75 

448.63130652 

581.044:30193 

750.05371848 

990.07642893 

1300.94867977 

76 

467.57002118 

608.19135822 

795.48040440 

1045..5.306.3252 

1380.00560055 

77 

487.27908003 

030.55996934 

830.20072402 

1104.0.3481731 

1463.80593658 

78 

507.77087347 

666.20516797 

879.07370085 

1105.7507.3220 

15.52.6.3429278 

79 

529.08170841 

097.184400.53 

924.02744889 

1230.87.335254 

164^.79235035 

80 

551.24497675 

729.55769854 

971.22882134 

1299.57138693 

1746.59989137 

81 

574.29477582 

763.38779497 

1020.79026240 

1372.04781321 

1852.39588485 

82 

598.26656085 

798.74024574 

1072.82977.5.52 

1448.51044293 

1964.53963794 

83 

623.19722952 

835.08355080 

1127.471264.30 

1529.17851729 

2083.41201622 

84 

649.12511870 

874.28931080 

1184.84482752 

1614.28.3.3.3574 

2209.41673719 

85 

676.09012345 

914.03233012 

1245.08700889 

1704.06891921 

2.342.98174142 

86 

704.13372839 

950.79071924 

1308.34142234 

1798.79270977 

2484.56064590 

87 

733.29907752 

1000.84037085 

1374.75849345 

1898.72630880 

2634.63428466 

88 

763.03104003 

1040.88446381 

1444.49041813 

2004.15625579 

279:3.712:34174 

89 

795.17028225 

1094.99420468 

1517.72123903 

2115.38484986 

2902.33508224 

90 

827.98333354 

1145.26900659 

1594.00730098 

22.32.73101660 

3141.07518718 

91 

862.10266688 

1197.80611188 

1075.33760003 

2356.53122251 

3330.53909841 

92 

897.58677356 

1252.70738692 

1700.104.549.34 

2487.14043975 

3531.37208032 

93 

934.49024450 

1310.07921933 

1849.10977080 

2624.93316394 

3744.25440513 

94 

972.80985428 

1370.03278420 

1942.56526564 

2770.30448796 

3909.90900944 

95 

1012.78464845 

14:32.68425949 

2040.69352892 

292:3.67123480 

4209.10424901 

96 

1054.29603439 

1498.15505117 

2143.72820537 

3085.47315271 

4462.65050459 

97 

1097.40787577 

1566.57202847 

2251.91461564 

3256.17417611 

4731.40953486 

98 

1142.36659090 

1038.00770975 

2365.51034042 

3430.26375579 

5016.29410095 

99 

1189.06125443 

1712.78081938 

2484.78586374 

3626.25826236 

5318.27175.337 

100 

1237.62370461 

1790.85595626 

2610.02515693 

3826.70246679 

5638.36805857 

23 


The  Present  Worth  of  Periodic  Payments  of  One 
Unit  Each. 


n 

1% 

a 

11 

i 

l%% 

2% 

1 

.99009901 

.98765432 

.98522167 

.98280098 

.98039216 

2 

1.97039506 

1.96311538 

1.95588342 

1.94869875 

1.94156094 

3 

2.94098521 

2.92653371 

2.91220042 

2.89798403 

2.88388327 

4 

3.90196555 

3.87805798 

3.854138465 

3.83094254 

3.80772870 

5 

4.85343124 

4.81783504 

4.78264497 

4.74785508 

4.71345951 

6 

5.79547647 

5.74600992 

5.69718717 

5.64899762 

5.6014:3089 

7 

6.72819453 

6.66272585 

6.59821396 

6.53464139 

6.47199107 

8 

7.65167775 

7.56812429 

7.48592508 

7.40505297 

7.32548144 

9 

8.56601758 

8.46234498 

8.36051732 

8.26049432 

8.16223671 

10 

9.47130453 

9.34552591 

9.22218455 

9.10122291 

8.98258501 

11 

10.36762825 

10.21780337 

10.07111779 

9.92719181 

9.78684805 

12 

11.25507747 

11.07931197 

10.90750521 

10.73954969 

10.57534122 

13 

12.13374007 

11.93018766 

11.73153222 

11.53764097 

11.34837375 

14 

13.00370304 

12.77055275 

12.54338150 

12.32200587 

12.10624877 

15 

13.86505252 

13.60054592 

13.34323301 

13.09288046 

12.849263.50 

16 

14.71787378 

14.42029227 

14.13126405 

13.85049677 

13.57770931 

17 

15.56225127 

15.22991829 

14.90764931 

14.59508282 

14.29187188 

18 

16.39826286 

16.02954893 

15.67256089 

15.32686272 

14.99203125 

19 

17.22600850 

16.81930759 

16.42616837 

16.04605673 

15.6784*201 

20 

18.04555297 

17.59931613 

17.16863878 

16.75288130 

16.35143334 

21 

18.85696313 

18.36969495 

17.90013673 

17.44754919 

17.01120916 

22 

19.66037934 

19.13056291 

18.62082437 

18.13026948 

17.65804820 

23 

20.45582113 

19.88203744 

19.33086145 

18.80124764 

18.29220411 

24 

21.24338726 

20.62423451 

20.03040537 

19.46068565 

18.91392560 

25 

22.02315570 

21.35726865 

20.71961120 

20.10878196 

19.52345647 

26 

22.79520366 

22.08125299 

21.39863172 

20.74573166 

20.12103576 

27 

23.55960759 

22.79629925 

22.06761946 

21.37172644 

20.70689780 

28 

24.31644316 

23.50251778 

22.72671671 

21.98695474 

21.28127236 

29 

25.06578530 

24.20001756 

23.37607558 

22.59160171 

21.84438466 

30 

25.80770822 

24.88890623 

24.01583801 

23.18584934 

22.39645555 

31 

26.54228537 

25.56929010 

24.64614582 

23.76987651 

22.93770152 

32 

27.2()958497 

26.24129417 

25.26713874 

24.34385897 

23.46833482 

33 

27.98969255 

26.90496215 

25.87895442 

24.90796951 

23.98856355 

34 

28.70266589 

27.56045644 

26.48172849 

25.46237789 

24.49859172 

35 

29.40858009 

28.20785821 

27.07559458 

26.00725100 

24.99861933 

36 

30.10750504 

28.84726737 

27.66068431 

26.54275283 

25.48884248 

37 

30.79950994 

29.47878259 

28.23712742 

27.06904455 

25.96945341 

38 

31.48466330 

30.10250132 

28.80505163 

27.58628457 

26.44064060 

39 

32.16303297 

30.71851982 

29.36458288 

28.09462857 

26.90258883 

40 

32.83468611 

31.32693316 

29.91584520 

28.59242955 

27.35547923 

41 

33.49968922 

31.92783522 

30.45896079 

29.08523789 

27.79948945 

42 

34.15810814 

32.52131874 

30.99405004 

29.56780136 

28.23479358 

43 

34.81000806 

33.10747529 

31.52123157 

30.04206522 

28.661562.33 

44 

35.45545352 

33.68639535 

32.04062223 

30.50817221 

29.07996307 

45 

36.09-450844 

34.25816825 

32.55233718 

30.96626261 

29.49015988 

46 

36.72723608 

34.82288222 

33.05648983 

31.41647431 

29.89231.360 

47 

37.35369909 

35.38062442 

33.55319195 

31.8.5894281 

.30.28658196 

48 

37.97395949 

35.93148096 

34.04255365 

32.29380129 

30.67311957 

49 

38.58807871 

36.47553670 

34.52468339 

32.72118063 

31.05207801 

50 

39.19611753 

37.01287575 

34.99968807 
24 

33.14120946 

31.42360589 

n 

1% 

IV4% 

IM.''/^ 

1%% 

2% 

51 

39.79813617 

37.54358099 

35.46767298 

33.55401421 

31.78784892 

52 

40.39419423 

38.06773431 

35.92874185 

33.95971913 

32.14494992 

53 

40.98535072 

38.58541660 

36.38299690 

34.3584t632 

32.49504894 

54 

41.5G866408 

39.09670776 

36.83053882 

34.75031579 

32.83828327 

55 

42.14719216 

39.60168667 

37.27146681 

35.13544550 

33.17478752 

56 

42.71999236 

40.10043128 

37.70587864 

35.51395135 

,33.50469365 

57 

43.38712102 

40.59301855 

38.13387058 

35.88594727 

33.82813103 

58 

43.84863468 

41.07952449 

38.55553751 

36.25154523 

34.14522655 

59 

44.404.58879 

41.56002419 

38.97097292 

36.61085526 

34.45610441 

60 

44.95503841 

42.03459179 

39.38026889 

36.96398552 

.34.76088667 

61 

45.50003803 

42.50330054 

39.78351615 

37.31104228 

35.05969282 

62 

46-.03964161 

42.96622275 

40.18080408 

37.65213000 

35..35264002 

63 

46.57390258 

43.42342988 

40.57222077 

37.98735134 

35.63984315 

64 

47.10287385 

43.87499247 

40.95785298 

38.31680723 

35.92141486 

65 

47.62660777 

44.32098022 

41.33778619 

38.64059678 

36.19746555 

66 

48.14515621 

44.76146194 

41.71210462 

38.95881748 

36.46810348 

67 

48.65857050 

45.19650562 

42.08089125 

39.27156509 

36.73343478 

68 

49.16690149 

45.62617839 

42.44422783 

39.57893375 

36.99.356.351 

69 

49.67019949 

46.05054656 

42.80219491 

39.88101597 

37.24859168 

70 

50.16851435 

46.46967561 

43.15487183 

40.17790267 

37.49861929 

71 

50.66189539 

46.88363023 

43.50233678 

40.46968321 

37.74374440 

72 

51.15039148 

47.29247430 

43.84466678 

40.75644.542 

37.984^6314 

73 

51.63405097 

47.69627091 

44.18193771 

41.0.3827560 

.38.21966975 

74 

52.11292175 

48.09508238 

44.51422434 

41.31.52.5857 

.38.4.5065661 

75 

52.58705124 

48.48897026 

44.84160034 

41.58747771 

.38.67711433 

76 

53.05648638 

48.87799532 

45.16413826 

41.85501495 

.38.89913169 

77 

53.52127364 

49.26221760 

45.48190962 

42.11795081 

39.11679578 

78 

53.98145905 

49.64169640 

45.79498485 

42.37636443 

39.33019194 

79 

54.43708817 

50.01649027 

46.10343335 

42.63033359 

39.53940386 

80 

54.88820611 

50.38665706 

46.40732349 

42.87993474 

39.74451359 

81 

55..3348574;3 

50.75225389 

46.70672265 

43.12524298 

39.94560156 

82 

55.77708666 

51.11333717 

47.00169720 

43..36633217 

40.14274662 

83 

56.21493729 

51.43996264 

47.29231251 

4:3.60327486 

40..33602610 

84 

56.64845276 

51.82218532 

47.57863301 

43.8.36142.37 

40.52551579 

85 

57.07767600 

52.17005958 

47.86072218 

44.06500479 

40.71128999 

86 

57.50264951 

52.51363909 

48.13864254 

44.28993099 

40.89342156 

87 

57.92441535 

52.85297688 

48.41245571 

44.51097869 

41.07198192 

88 

58.34001520 

53.18812531 

48.68222237 

44.72824441 

41.24704110 

89 

58.75249030 

53.51913611 

48.94800233 

44.94176.355 

41.41866774 

90 

59.16088148 

53.84606035 

49.20985452 

45.15161037 

41.58692916 

91 

59.56522919 

54.16894850 

49.46783696 

45.35784803 

41.75189133 

92 

59.96557346 

54.48785037 

49.72200686 

45.56053860 

41.91361895 

93 

60.36195392 

54.80281518 

49.97242055 

45.75974310 

42.07217544 

94 

60.75440982 

55.11389154 

50.21913355 

45.95.5.52147 

42.22762299 

95 

61.14298002 

55.42112744 

50.46220054 

46.14793265 

42.38002254 

96 

61.52770299 

55.72457031 

50.70167541 

46.33703455 

42.52943386 

97 

61.90861682 

56.02426698 

50.93761124 

46.52288408 

42.67591555 

98 

62.28575923 

56.32026368 

51.17006034 

46.70553718 

42.819.52.505 

99 

62.65916755 

56.61260610 

51.39907422 

46.88504882 

42.96031867 

100 

63.02887877 

56.90133936 

51.62470367 

47.06147304 

43.09835164 

25 


n 

21/4% 

21/2% 

2%% 

3% 

31/2% 

1 

.97799551 

.97560976 

.97323601 

.97087379 

.96618357 

2 

1.934+6755 

1.92742415 

1.920424:34 

1.91.346970 

1.89969428 

3 

2.86989()87 

2.85602356 

2.84226213 

2.82861135 

2.8016:3698 

4 

3.784.74021 

3.76197421 

3.73942787 

3.71709840 

3.67307921 

5 

4.6794^253 

4.64582850 

4.61258186 

4.57970719 

4.51505238 

6 

5.55447680 

5.50812536 

5.46236678 

5.41719144 

5.32855302 

7 

C.41024626 

6.34939060 

6.28940806 

6.23028296 

6.114.54398 

8 

7.247184-61 

7.17013717 

7.09431441 

7.01969219 

6.87395554 

9 

8.06570622 

7.97086553 

7.87767826 

7.78610892 

7.60768651 

10 

8.86621635 

8.75206393 

8.64007616 

8.53020284 

8.31660532 

11 

9.64911134 

9.51420871 

9.38206926 

9.25262911 

9.00155104 

12 

10.41477882 

10.25776460 

10.10420.366 

9.95400400 

9.663334:33 

13 

11.16359787 

10.98318497 

10.80701086 

10.634955.33 

10.30273849 

14 

11.89593924 

11.69091217 

11.49100814 

11.29607314 

10.92052028 

15 

12.61216551 

12.38137773 

12.15669892 

11.93793509 

11.51741089 

16 

13.31263131 

13.05500266 

112.80457315 

12.56110203 

12.09411681 

17 

13.99768343 

13.71219772 

13.43510769 

13.16611847 

12.65132059 

18 

14.66766106 

14.35336363 

14.04876661 

13.75351308 

13.18968173 

19 

15.32289590 

14.97889134 

14.64600157 

14.32379911 

13.70983742 

20 

15.96371237 

15.58916229 

15.22725213 

14.87747486 

14.21240330 

21 

16.59042775 

16.184.54857 

15.79294611 

15.41502414 

14.69797420 

22 

17.20335232 

16.76541324 

16.34349987 

15.93691664 

15.16712483 

23 

17.80278955 

17.33211048 

16.87931861 

16.44360839 

15.62041047 

24 

18.38903624 

17.88498583 

17.40079670 

16.93554212 

16.05836760 

25 

18.96238263 

18.42437642 

17.90831795 

i  7.41314769 

16.48151459 

26 

19.52311260 

18.95061114 

18.40225592 

17.87684242 

16.89036226 

27 

20.07150376 

19.46401087 

18.88297413 

18.32703147 

17.28536451 

28 

20.60782764 

19.96488866 

19.35082640 

18.76410823 

17.66701885 

29 

21.13234977 

20.45354991 

19.80615708 

19.18845459 

18.03576700 

30 

21.64532985 

20.93029259 

20.24930130 

19.60044135 

18.39204541 

31 

22.14702186 

21.39540741 

20.68058520 

20.00042849 

18.73627576 

32 

22.63767419 

21.84917796 

21.10032623 

20.38876553 

19.06886.547 

33 

23.117.52977 

22.29188094 

21.508833.32 

20.76579177 

19.39020818 

34 

23.58682618 

22.72378628 

21.90640712 

21.13183667 

19.70068423 

35 

24.04579577 

23.14515734 

22.29334026 

21.48722007 

20.00066110 

36 

24.4946(i579 

23.55625107 

22.66991753 

21.83225249 

20.29049:381 

37 

24.9336581.8 

2.3.95731811 

23.()3()11609 

22.1672354;} 

2().57()5251-2 

38 

25.36299118 

24.34860304 

23..39310568 

22.49246158 

20.841087:36 

39 

25.78287646 

24.7303444;? 

23.74024884 

22.80821513 

21.10249987 

40 

26.19352221 

25.10277505 

24.07810102 

2.3.11477197 

21.35507234 

41 

26.59513174 

25.46612200 

24.40691101 

23.412:39997 

21. .599 10371 

42 

26.98790390 

25.820()0683 

24.72692069 

2:3.701:35920 

21.83488281 

43 

27.37203316 

2().l(i(iMr)(i9 

25.()3S3(i563 

23.98190213 

22.()()268870 

44 

27.74770969 

2(i.50384.945 

35.34147.507 

24.251.27392 

22.28279102 

45 

28.11511950 

26.83302386 

25.63647209 

24.51871254 

22.49545026 

46 

2S.471. 1-4 1-50 

27.15416962 

25.92357.381 

24.77544907 

22.7()091HI2 

47 

2K.H2.1H6259 

27.46748255 

26.20299151. 

25.02470783 

22.8994:5780 

48 

2!>. 16954777 

27.77315371 

26.47493094 

25.26670664 

2:5.09 1241-25 

49 

29..50567019 

28.07136947 

26.73959215 

25.50165693 

2;}.276561..50 

50 

29.83439627 

28.36231168 

26.99716998 

25.72976401 

23.45561787 

26 


n 

2y4% 

^y2% 

2%% 

3% 

3yo% 

51 

30.15588877 

28.64615774 

27.24785399 

25.95122719 

23.628616-30 

52 

30.47030fi87 

28.92308072 

27.49182870 

26.16623999 

23.79576455 

53 

30.77780()2;} 

29.19324948 

27.72927368 

26.37499028 

23.957260U 

54 

31.07853910 

29.45682876 

27.96036368 

26.57766047 

24.11329511 

55 

31.37265438 

29.71397928 

28.18526879 

26.77442764 

24.26405-324 

56 

31.(56029768 

29.96485784 

28.40415454 

26.96346373 

24.40971327 

57 

31.94.16114.2 

30.20961740 

28.61718203 

27.15093566 

24.55044761 

58 

32.21673489 

30.44840722 

28.82450806 

27.33100549 

24.68642281 

59 

32.48580429 

30.68137290 

29.02628522 

27.50583058 

24.81779981 

60 

32.74895285 

30.90865649 

29.22266201 

27.67556367 

24.94473412 

61 

33.00631086 

31.13039657 

29.41378298 

27.8403.5.307 

25.06737596 

62 

33.25800573 

31.34672836 

29.59978879 

28.00034279 

25.18-587050 

63 

33.50416208 

31.55778377 

29.78081634 

28.15567261 

25.-30035797 

64 

33.74490179 

31.76369148 

29.95699887 

28.30647826 

25.41097389 

65 

33.98034405 

31.96457705 

30.1284()605 

28.45289152 

25.51784916 

66 

34.21060543 

32.16056298 

30.295.34409 

28.59504031 

25.62111030 

67 

34.43579993 

32.35176876 

30.45775.581 

28.7-3:304884 

25.72087952 

68 

34.65603905 

32.53831099 

30.61582074 

28.86703771 

25.81727489 

69 

34.87143183 

32.72030340 

.30.76965522 

28.99712-399 

25.91041053 

70 

35.08208492 

32.89795698 

30.91937247 

29.12:342135 

26.00039664 

71 

35.28810261 

33.07107998 

31.06508270 

29.24604105 

26.087.33976 

72 

35.48958691 

33.24007803 

31.20689304 

29.365087.52 

26.17134276 

73 

35.68663756 

33.40495417 

31. .344908 16 

29.48066750 

26.25250508 

74 

35.87935214 

33.56580895 

31.47922936 

29.59288107 

26.33092278 

75 

36.06782605 

33.72274044 

31.609955.58 

29.70182628 

26.40668868 

76 

36.25215262 

33.875844:33 

31.73718,304 

29.807598.33 

26.47989244 

77 

36.43242310 

34.02521398 

31.86100540 

29.91028964 

26.55062072 

78 

36.60872675 

34.17094047 

31.98151377 

30.00998994 

26.61895721 

79 

36.78115085 

34.31311265 

.32.09879685 

30.10678635 

26.68498281 

80 

36.94978079 

34.45181722 

32.21294098 

30.20076345 

26.74877567 

81 

37.11470004 

34.58718752 

.32..32403015 

30.29200335 

26.81041127 

82 

37.27599026 

34.71915976 

.32.4:3214613 

30.38058577 

26.86996258 

83 

37.43373130 

34.84796074 

:32..53736850 

.30.46658813 

26.92750008 

84 

37.58800127 

34.97362023 

32.63977469 

.30.55008556 

26.98:309186 

85 

37.73887655 

35.09621486 

32.73944009 

30.63115103 

27.03680373 

86 

37.88643183 

35.21581938 

32.83643804 

30.70985537 

27.08869926 

87 

38.03074018 

35..33250671 

32.93083995 

30.78626735 

27.1:3883986 

88 

38.17187304 

35.44634891 

33.02271528 

30.86045374 

27.18728489 

89 

38.30990028 

35.55741269 

33.11213165 

.30.93247935 

27.2:3409168 

90 

38.44489025 

35.66576848 

.33.19915489 

31.00240714 

27.27931564 

91 

38.57690978 

35.77148144 

33.28384905 

31.07029820 

27.-32:301028 

92 

38.70602423 

35.87461604 

33.36627644 

31.1:3621184 

27.36522732 

93 

38.83229754 

35.97523516 

33.44649776 

31.20020567 

27.40601673 

94 

38.9.5579221 

36.073400  Ki 

.33.-52457203 

31.262:]35(;0 

27.44542630 

95 

39.07657940 

.36.16917089 

33.60055672 

31.;i22U5592 

27.48350415 

96 

39.19468890 

36.26260574 

33.67450775 

31.-381219-34 

27.52029386 

97 

39.31020920 

36.35376170 

33.74647957 

31.43807703 

27.55583948 

98 

39.42318748 

36.41269434 

33.81652512 

31.49:327867 

27.59018307 

99 

39.5;3;367968 

36.52945790 

33.88469599 

31.54687250 

27.62336529 

too 

39,64174052 

36.61410526 

33.95104232 

31,59890534 

27.65542540 

27 


n 

*% 

*V2% 

5% 

5V2% 

6% 

1 

.96153846 

.95693780 

.95238095 

.94786730 

.94339623 

2 

1.88609467 

1.87266775 

1.85941043 

1.84631971 

1.83339267 

3 

2.77509103 

2.74896435 

2.72324803 

2.69793338 

2.67301195 

4 

3.62989522 

3.58752570 

3.54595050 

3.50515012 

3.46510561 

S 

4.45182233 

4.38997674 

4.32947667 

4.27028448 

4.21236379 

6 

6.24213686 

5.1,5787248 

5.07569207 

4.99553031 

4.91732433 

7 

6.00205467 

5.89270094 

5.78637340 

5.68296712 

5.58238144. 

8 

6.73274487 

6.59588607 

6.46321276 

6.33456599 

6.20979381 

9 

7.43533161 

7.26879050 

7.10787168 

6.95219525 

6.80169227 

10 

8.11089578 

7.91271818 

7.72173493 

7.53762583 

7.36008705 

11 

8.76047671 

8.52891692 

8.30641422 

8.09253633 

7.88687458 

12 

9.38507376 

9.11858078 

8.86325164 

8.61851785 

8.38384394 

13 

9.98564785 

9.68285242 

9.39357299 

9.11707853 

8.85268296 

14 

10.56312293 

10.22282528 

9.89864094 

9.58964790 

9.29498393 

15 

11.11838743 

10.73954573 

10.37965804 

10.03758094 

9.71224899 

16 

11.65229561 

11.23401505 

10.83776956 

10.46216203 

10.10589527 

17 

12.16566885 

11.70719143 

11.27406625 

10.86460856 

10.47725969 

18 

12.65929698 

12.15999180 

11.68958690 

11.24607447 

10.82760348 

19 

13.13393940 

12.59329359 

12.08532086 

11.60765352 

11.15811649 

20 

13.59032634 

13.00793645 

12.46221034 

11.95038248 

11.46992122 

21 

14.02915995 

13.40472388 

12.82115271 

12.27524406 

11.76407662 

22 

14.45111533 

13.78442476 

13.16300258 

12.58316973 

12.04158172 

23 

14.85684167 

14.14777489 

13.48857388 

12.87504239 

12.30007898 

24 

15.24696314 

14.49547837 

13.79864179 

13.15169895 

12.55035753 

25 

15.62207994 

14.82820896 

14.09394457 

13.41393266 

12.78335616 

26 

15.98276918 

15.14661145 

14.37518530 

13.66249541 

13.00316619 

27 

16.32958575 

15.45130282 

14.64303362 

13.89809991 

13.21053414 

28 

16.66306322 

15.74287351 

14.89812726 

14.12142172 

13.40616428 

29 

16.98371463 

16.02188853 

15.14107358 

14.33310116 

13.59072102 

30 

17.29203330 

16.28888854 

15.37245103 

14.53374517 

13.76483115 

31 

17.58849356 

16.54439095 

15.59281050 

14.72392907 

13.92908600 

32 

17.87355150 

16.78889086 

15.80267667 

14.90419817 

1408404339 

33 

18.14764567 

17.02286207 

16.00254921 

15.07506936 

14.23022961 

34 

18.41119776 

17.24675796 

16.19290401 

15.23703257 

14.36814114 

35 

18.66461323 

17,46101240 

16.37419429 

15.39055220 

14.49824636 

36 

18.90828195 

17.66604058 

16.54685171 

15.53606843 

14.62098713 

37 

19.14257880 

17.86223979 

16.71128734 

15.67399851 

14.73678031 

38 

19.36786423 

18.04999023 

16.86789271 

15.80473793 

14.84601916 

39 

19.58448484 

18.22965572 

17.01704067 

15.92866154 

14.94907468 

40 

19.79277388 

18.40158442 

17.15908635 

16.04612469 

15.04629687 

41 

19.99305181 

18.56610949 

17.29436796 

16.15746416 

15.13801592 

42 

20.18562674 

18.72354975 

17.42320758 

16.26299920 

15.22454332 

43 

20.37079494 

18.87421029 

17.54591198 

16.36303242 

15.30617294 

44 

20.54884129 

19.01838305 

17.66277331 

16.45785063 

15.38318202 

45 

20.72003970 

19.15634742 

17.77406982 

16.54772572 

15.45583209 

46 

20.88465356 

19.28837074 

17.88006650 

16.63291537 

15.52436990 

47 

21.04293612 

19.41470884 

17.98101571 

16.71366386 

15.58902821 

48 

21.19513088 

19.535()0654 

18.07715782 

16.79020271 

15.65002661 

49 

21.34147200 

19.65129813 

18.16872173 

16.86275139 

15.70757227 

50 

21.48218462 

19.76200778 

18.25592546 

16.93151790 

15.76186064 

28 


n 

■i% 

•*y.% 

5% 

5^2% 

6% 

51 

21.6174.8521 

19.86795003 

18.33897663 

16.99669943 

15.81306707 

52 

21.74758193 

19.96933017 

18.41807298 

17.05848287 

15.861392.52 

53 

21.872()-71.;):3 

20.06634466 

18.493K)284 

17.11704.538 

15.90697408 

54 

21.992J)5(i()7 

20.15918149 

18.56514.556 

17.172.55486 

15.94997544 

55 

22.108()1218 

20.24802057 

18.63347196 

17.22517048 

15.99054297 

56 

22.21981940 

20.33303404 

18.69854473 

17.27504.311 

16.02881412 

57 

22.;32G74943 

20.41438664 

18.76951879 

17.32231575 

16.06491898 

58 

22.42956676 

20.49223602 

18.81954170 

17.36712393 

16.09898017 

59 

22.52852957 

20.56673303 

18.87575400 

17.40959614 

16.131113.37 

60 

22.62348977 

20.63802204 

18.92928953 

17.44985416 

16.16142771 

61 

22.71489421 

20.70624118 

18.98027574 

17.48801343 

16.19002614 

62 

22.80278289 

20.77152266 

19.02883404 

17.524183.34 

16.21700579 

63 

22.88729124 

20.83399298 

19.07508003 

17.55846762 

16.24245829 

64 

22.96854927 

20.89377319 

19.11912384 

17.59096457 

16.26647009 

65 

23.04668199 

20.95097913 

19.16107033 

17.621767.37 

16.28912272 

66 

23.12180961 

21.00572165 

19.20101936 

17.65096433 

16.31049314 

67 

23.19404770 

21.05810684 

19.23906606 

17.6786-3917 

16. .3306-5390 

68 

23.26350740 

21.10823621 

19.27530101 

17.70487125 

16.34967.349 

69 

23.33029558 

21.15620690 

19.30981048 

17.72973579 

16.-367616-50 

70 

23.39451498 

21.20211187 

19.34267665 

17.75330406 

16.38454.387 

71 

23.45626440 

21.24604007 

19.37397776 

17.77564366 

16.40051308 

72 

23.51563885 

21.28807662 

19.40378834 

17.79681864 

16.41557838 

73 

23.57272966 

21.32830298 

19.43217937 

17.81688970 

16.42979093 

74 

23.62762468 

21.36679711 

19.45921845 

17.83591441 

16.44319899 

75 

23.68040834 

21.40363360 

19.48496995 

17.85394731 

16.45584810 

76 

23.73116187 

21.43888.383 

19.50949519 

17.87104010 

16.46778123 

77 

23.77996333 

21.47261611 

19.53285257 

17.88724180 

16.47903889 

78 

23.82688782 

21.50489579 

19.55.509768 

17.90259887 

16.48965933 

79 

23.87200752 

21.53578545 

19.57628351 

17.917155.32 

16.49967862 

80 

23.91539185 

21.56534493 

19.59646048 

17.93095291 

16.50913077 

81 

23.95710755 

21.59363151 

19.61567665 

17.94403120 

16.51804790 

82 

23.99721879 

21.62070001 

19.63397776 

17.95642767 

16.52646028 

83 

24.03578730 

21.64660288 

19.65140739 

17.96817789 

16.53439649 

84 

24.07287241 

21.671390.32 

19.66800704 

17.97931554 

16.54188348 

85 

24.10853116 

21.69511035 

19.68381623 

17.98987255 

16.54894668 

86 

24.14281842 

21.71780895 

19.69887260 

17.99987919 

16.55561008 

87 

24.17578694 

21.73953009 

19.71321200 

18.00936416 

16.56189630 

88 

24.20748745 

21.76031588 

19.72686857 

18.01835466 

16.56782670 

89 

24.23796870 

21.78020658 

19.73987483 

18.02687645 

16.57342141 

90 

24.26727759 

21.79924075 

19.75226174 

18.03495398 

16.57869944 

91 

24.29545923 

21.8174.5526 

19.76405880 

18.04261041 

16.58367872 

92 

24.32255695 

21.83488542 

19.77529410 

18.04986769 

16.58837615 

93 

24.34861245 

21.85156499 

19.785994:38 

18.05674662 

16.59280769 

94 

24.37366582 

21.86752631 

19.79618512 

18.06326694 

16.59698839 

95 

24.39775559 

21.88280030 

19.80589059 

18.06944734 

16.60093244 

96 

24.42091884 

21.897416.55 

19.81513390 

18.07530.553 

16.60465325 

97 

24.44319119 

21.91140340 

19.82393705 

18.08085832 

16.60816344 

98 

24.46460692 

21.92478794 

19.83232100 

18.08612164 

16.61147494 

99 

24.48519896 

21.93759612 

19.840.30571 

18.09111055 

16.61459900 

100 

24.50499900 

21.94985274 

19.84791020 

18.09583939 

16.61754623 

29 


The  Periodic  Payment  That  Will  Amount  to  One. 
The  Sinking  Fund. 


Sn 

S°— 1 

n 

1% 

lVi% 

iy2% 

1%% 

2% 

1 

1.00 

1.00 

1.00 

1.00 

1.00 

2 

.49751244 

.49689441 

.49627792 

.49566295 

.49504950 

3 

.33002211 

.32920117 

.32838296 

.32756746 

.32675467 

4 

.26428109 

.24536102 

.24444478 

.24353237 

.24262375 

5 

.19G03980 

.19506211 

.19408932 

.19312142 

.19215839 

6 

.16254837 

.16153381 

.16052521 

.15952256 

.15852581 

7 

.13862828 

.13758872 

.13655616 

.13553059 

.13451196 

8 

.12069029 

.11963314 

.11959402 

.11754292 

.11650980 

9 

.10674036 

.10567055 

.10460982 

.10355813 

.10251544 

10 

.09558208 

.09450307 

.09343418 

.09237534 

.09132653 

11 

.08645408 

.08536839 

.08429384 

.08323038 

.08217794 

12 

.07884879 

.07775831 

.07667999 

.07561337 

.07455960 

13 

.07241482 

.07132100 

.07024036 

.06917283 

.06811835 

14 

.06690177 

.06580515 

.06472332 

.06365562 

.06260197 

15 

.06212378 

.06102646 

.05994436 

.05887739 

.05782547 

16 

.05794460 

.05684672 

.05576508 

.05469958 

.05365013 

17 

.05425806 

.05316023 

.05207966 

.05101623 

.04996984 

18 

.05098205 

.04988479 

.04880578 

.04774492 

.04670210 

19 

.04805175 

.04695548 

.04587847 

.04482061 

.04378177 

20 

.04541531 

.04432039 

.04:324574 

.04219122 

.04115672 

21 

.04303075 

.04193748 

.04086550 

.03981464 

.03878477 

22 

.04086372 

.03977238 

.03870331 

.03765638 

.03663140 

23 

.03888584 

.03779666 

.03673075 

.03568796 

.03466810 

24 

.03707347 

.03598665 

.03492410 

.03388565 

.03287110 

25 

.03540675 

.03432247 

.03326345 

.03222952 

.03122044 

26 

.03386888 

.03278729 

.03173196 

.03070269 

.02969923 

27 

.03244553 

.03136677 

.03031527 

.02929079 

.02829309 

28 

.03112444 

.03004863 

.02900108 

.02798151 

.02698967 

29 

.02989502 

.02882228 

.02777878 

.02676424 

.02577836 

30 

.02874811 

.02767854 

.02663919 

.02562975 

.02464992 

31 

.02767573 

.02660942 

.02557430 

.02457005 

.02359635 

32 

.02667089 

.02560791 

.02457710 

.02357812 

.02261061 

33 

.02572744 

.02466786 

.02364144 

.022()4779 

.02168653 

34 

.02483997 

.02378387 

.02276189 

.02177363 

.02081867 

35 

.02400368 

.02295111 

.02193363 

.02095082 

.02000221 

36 

.02321431 

.02216533 

.02115240 

.02017507 

.01923285 

37 

.02246805 

.02142270 

.02041 4:}7 

.01944257 

.01850678 

38 

.02176150 

.02071983 

.01971613 

.01874990 

.01782057 

39 

.02109160 

.020053()5 

.01905463 

.01809399 

.01717114 

40 

.02045560 

.01942141 

.01842710 

.01747209 

.01655575 

41 

.01985102 

.01882063 

.01783106 

.01688170 

.01597188 

42 

.01927563 

.01824906 

.01726426 

.01632057 

.01541729 

43 

.01872737 

.01770466 

.01672465 

.01578666 

.01488993 

44 

.01820441 

.01718557 

.01621038 

.01527810 

.01438794 

45 

.01770505 

.01669012 

.01571976 

.01479321 

.01390962 

30 


n 

1% 

IV4% 

1^3%, 

iy4% 

2% 

46 

.01722775 

.01621675 

.01525125 

.01433043 

.01345342 

47 

.01677111 

.01576406 

.01480342 

.01388836 

.01301792 

48 

.0ie38384 

.01533075 

.01437500 

.01346569 

.01260184 

49 

.01591474. 

.01491563 

.01396478 

.01306124 

.01220396 

50 

.01551273 

.01451763 

.01357168 

.01267391 

.01182321 

51 

.01512680 

.01413571 

.01319469 

.01230269 

.01145856 

52 

.01475603 

.01376897 

.01283287 

.01194665 

.01110909 

53 

.01439956 

.013416,53 

.01248537 

.01160492 

.01077392 

54 

.01405658 

.01307760 

.01215138 

.01127672 

.01045226 

55 

.01372637 

.01275145 

.01183018 

.01096129 

.01014337 

56 

.01340824 

.01243739 

.01152106 

.01065795 

.00984656 

57 

.01310156 

.01213478 

.01122341 

.01036606 

.00956120 

58 

.01280573 

.01184303 

.01093661 

.01008503 

.00928667 

59 

.01252020 

.01156158 

.01066012 

.00981430 

.00902243 

60 

.01224445 

.01128993 

.01039343 

.00955336 

.00876797 

61 

.01197800 

.01102758 

.01013604 

.00930172 

.00852278 

62 

.01172041 

.01077410 

.00988751 

.00905892 

.00828643 

63 

.01147125 

.01052904 

.00964741 

.00882455 

.00805848 

64 

.01123013 

.01029203 

.00941534 

.00859821 

.00783855 

65 

.01099667 

.01006268 

.00919094 

.00837952 

.00762624 

66 

.01077052 

.00984065 

.00897386 

.00816813 

.00742122 

67 

01055136 

.00962560 

.00876376 

.00796372 

.00722316 

68 

.01033889 

.00941724 

.00856033 

.00776597 

.00703173 

69 

.01013280 

.00921527 

.00836329 

.00757459 

.00684665 

70 

.00993282 

.00901941 

.00817235 

.00738930 

.00666765 

71 

.00973870 

.00882941 

.00798727 

.00720985 

.00649446 

72 

.00955019 

.00864501 

.00780779 

.00703600 

.00632683 

73 

.00936706 

.00846600 

.00763368 

.00686750 

.00616454 

74 

.00918910 

.00829215 

.00746473 

.00670413 

.00600736 

75 

.00901609 

.00812325 

.00730072 

.00654570 

.00585508 

76 

.00884784 

.00795910 

.00714146 

.00639200 

.00570751 

77 

.00868416 

.00779953 

.00698676 

.00624285 

.00556447 

78 

.00852488 

.00764436 

.00683645 

.00609806 

.00542576 

79 

.00836983 

.00749341 

.00669036 

.00595748 

.00529123 

80 

.00821885 

.00734652 

.00654832 

.00582093 

.00516071 

81 

.00807179 

.00720356 

.00641019 

.00568828 

.00503405 

82 

.00792851 

.007064;37 

.00627583 

.00555936 

.00491110 

83 

.00778887 

.00692881 

.00614509 

.00543406 

.00479173 

84 

.00765273 

.00679675 

.00601784 

.00531223 

.00467581 

85 

.00751988 

.00666808 

.00589396 

.00519375 

.00456321 

86 

.00739050 

.00654267 

.00577333 

.00507850 

.00445381 

87 

.00726418 

.00642041 

.00565584 

.00496636 

.00434750 

88 

.00714089 

.00630119 

.00554138 

.00485724 

.00424416 

89 

.00702056 

.00618491 

.00542984 

.00475102 

.00414370 

90 

.00690306 

.00607146 

.00532113 

.00464760 

.00404602 

91 

.00678832 

.00596076 

.00521516 

.00454690 

.00395101 

92 

.00667624 

.00585272 

.00511182 

.00444882 

.00385859 

93 

.00656673 

.00574724 

.00501104 

.00435327 

.00376868 

94 

.00645971 

.00564425 

.00491273 

.00426017 

.00368118 

95 

.00635511 

.00554366 

.00481681 

.00416944 

.00359602 

96 

.00625284 

.00544541 

.00472321 

.00408101 

.00351313 

97 

.00615284 

.00534941 

.00463186 

.00399480 

.00341242 

98 

.00605503 

.00525560 

.00454268 

.00391074 

.00335383 

99 

.00595936 

.00516391 

.00455560 

.00382876 

.00327729 

100 

.00586574 

.00507428 

.00437057 

.00374880 

.00320274 

31 


n 

21/4% 

2y3% 

2%% 

3% 

3yo% 

1 

1.00 

1.00 

1.00 

1.00 

1.00 

2 

.494.«758 

.49382716 

.49321825 

.49261048 

.49140049 

3 

.32591.1-58 

.32513717 

.3243324:3 

.32353036 

.32193418 

4 

.24171893 

.24081788 

.23992059 

.23902705 

.23725114 

5 

.19120021 

.19024686 

.18929832 

.18835457 

.18648137 

6 

.157.53496 

.15654997 

.15557083 

.15459750 

.15266821 

7 

.13350026 

.13249543 

.13149747 

.13050635 

.12854449 

8 

.11548462 

.11446735 

.11344795 

.1124.5639 

.11047665 

9 

.10148170 

.10045689 

.09944095 

.09843386 

.09644601 

10 

.09028768 

.08925876 

.08823972 

.08723051 

.08524137 

11 

.08113649 

.08010596 

.07908629 

.07807745 

.07609197 

12 

.07351740 

.07248713 

.07146871 

.07046209 

.06848395 

13 

.06707686 

.06604827 

.06503252 

.06402954 

.06206157 

14 

.06156230 

.06053653 

.05952457 

.05852634 

.05657073 

15 

.05678852 

.05576646 

.05475917 

.05376658 

.05182507 

16 

.05261663 

.05159899 

.0.5059710 

.04961085 

.04768483 

17 

.04894039 

.04792777 

.04693186 

.04595253 

.04404313 

18 

.04567720 

.04467008 

.04368063 

.04270870 

.04081684 

19 

.04276182 

.04176062 

.04077802 

.03981388 

.03794033 

20 

.04014207 

.03914713 

.03817178 

.03721571 

.03536108 

21 

.03777572 

.036787.33 

.03581941 

.03487178 

.03303659 

22 

.03562821 

.03464661 

.03368640 

.03274739 

.03093207 

23 

.03367097 

.03269638 

.03174410 

.03081390 

.02901880 

24 

.03188023 

.03091282 

.02996863 

.02904742 

.02727283 

25 

.03023599 

.02927592 

.02833997 

.02742787 

.02567404 

26 

.02872134 

.02776875 

.02684116 

.02592829 

.02420540 

27 

.02732188 

.02637687 

.02545776 

.02456421 

.02285241 

28 

.02602525 

.02508793 

.02417738 

.02329323 

.02160265 

29 

.02482081 

.02389127 

.02298935 

.02211467 

.02044538 

30 

.02369934 

.02277764 

.02188442 

.02101926 

.01937133 

31 

.02265280 

.02173900 

.02085453 

.01999893 

.01837240 

32 

.02167415 

.02076831 

.01989263 

.01904662 

.01744150 

33 

.02075722 

.019859.38 

.01899253 

.01815612 

.01657242 

34 

.01989655 

.01900675 

.01814875 

.01732196 

.01575966 

35 

.01908731 

.01820558 

.01735645 

.01653929 

.01499835 

36 

.018.32522 

.01745158 

.016611.32 

.01580379 

.01428416 

37 

.01760643 

.01674090 

.01590953 

.01511162 

.01361325 

38 

.01692753 

.01607012 

.01524764 

.01445934 

.01298214 

39 

.0162H51.;3 

.01543615 

.01462256 

.01384385 

.01238775 

40 

.01567738 

.01483623 

.01403151 

.01326238 

.01182728 

41 

.01510087 

.01426786 

.01347200 

.01271241 

.01129822 

42 

.01455364 

.01372876 

.01294175 

.01219167 

.01079828 

43 

.01403.364 

.01321688 

.01243871 

.01169811 

.01032.539 

44 

.01353901 

.01273037 

.01196100 

.01122985 

.00987768 

45 

.01306805 

.01226752 

.011,50693 

.01078518 

.009453443 

46 

.01261921 

.01182676 

.01107493 

.01036254 

.00905108 

47 

.01219107 

.01140669 

.01066358 

.00996051 

.00866916 

48 

.01178233 

.01100599 

.01027158 

.00957777 

.00830646 

49 

.01139179 

.01062348 

.00989773 

.00921314 

.00796167 

50 

.01101836 

.01025806 

.00954092 

.00886660 

.00763371 

32 


n 

2y4% 

2^% 

2%% 

3% 

^Vz% 

51 

.010()6102 

.00990870 

.00920014 

.00853382 

.00732156 

52 

.01031884. 

.00957446 

.00887M4 

.00821718 

.00702429 

53 

.00!)99091. 

.00925449 

.00856297 

.00791471 

.00674100 

54 

.009G7()51. 

.00894799 

.00826491 

.00762558 

.00647090 

55 

.00937489 

.00865419 

.00797953 

.00734907 

.00621323 

56 

.00908530 

.00837243 

.00770612 

.00708447 

.00596730 

57 

.0088071:i 

.00810204 

.00744404 

.00683114 

.00573245 

58 

.00853977 

.007842U 

.00719270 

.00658848 

.00550810 

59 

.00828268 

.00759307 

.00695153 

.00635593 

.00529366 

60 

.00803533 

.00735340 

.00672002 

.00613296 

.00508862 

61 

.00779724 

.00712294 

.00649767 

.00591908 

.00489249 

62 

.0075()795 

.00690126 

.00628402 

.00571385 

.00470480 

63 

.00734704 

.00668790 

.00607866 

.00551682 

.00452513 

64 

.00713411 

.00648249 

.00588118 

.00532760 

.00435308 

65 

.00692878 

.00628463 

.00569120 

.00514581 

.00418826 

66 

.00673070 

.00609398 

.00550837 

.00497110 

.00403031 

67 

.00653955 

.00591021 

.00533236 

.00480313 

.00387892 

68 

.00635500 

.00573300 

.00516285 

.00464159 

.00373375 

69 

.00617677 

.00556206 

.00499955 

.00448618 

.003594.53 

70 

.00600458 

.00539712 

.00484218 

.00433663 

.00346095 

71 

.00583816 

.00523790 

.00469048 

.00419266 

.00333277 

72 

.00567728 

.00508417 

.00454420 

.00405404 

.00320973 

73 

.00552169 

.00493568 

.00440311 

.00392053 

.00309160 

74 

.00537118 

.00479222 

.00426698 

.00379191 

.00297816 

75 

.00522554 

.00465358 

.00413560 

.00366796 

.00286919 

76 

.00508457 

.00451956 

.00400878 

.00354849 

.00276450 

77 

.00494808 

.004438997 

.00388633 

.00343331 

.00266390 

78 

.00481589 

.00426468 

.00376806 

.00332224 

.00256721 

79 

.00468784 

.00414338 

.00365382 

.00321510 

.00247426 

80 

.00456376 

.00402605 

.00354342 

.00811175 

.00238489 

81 

.00444850 

.00391248 

.00.343674 

.00301201 

.00229894 

82 

.00432692 

.00380254 

.00333361 

.00291576 

.00221628 

83 

.00421387 

.00369608 

.00323389 

.00282284 

.00213676 

84 

.00410423 

.00359298 

.00313747 

.00273313 

.00206025 

85 

.00399787 

.00349810 

.00304420 

.00264650 

.00198662 

86 

.00389467 

.00339633 

.00295397 

.00256284 

.00191576 

87 

.00379452 

.00330255 

.00286667 

.00248202 

.00184756 

88 

.00369730 

.00321165 

.00278219 

.00240393 

.00178190 

89 

.00360291 

.00312353 

.00270041 

.00232848 

.00171868 

90 

.00351126 

.00.303809 

.00262125 

.00225556 

.00165781 

91 

.00342224 

.00295523 

.00254460 

.00218508 

.00159919 

92 

.003.33577 

.00287486 

.00247038 

.00211694 

.00154273 

93 

.00325176 

.00279690 

.00239850 

.00205107 

.00148834 

94 

.00317012 

.00272126 

.00232887 

.00198737 

.00143591 

95 

.00309078 

.00264786 

.00226141 

.00192577 

.00138546 

96 

.00.301366 

.00257662 

.00219605 

.00186619 

.00133682 

97 

.00293868 

.00250747 

.00213272 

.00180856 

.00128995 

98 

.00286578 

.00244034 

.00207134 

.00175281 

.00124478 

99 

.00279489 

.00237517 

.00201185 

.00169886 

.00120124 

100 

.00272594 

.00231188 

.00195418 

.00164667 

.00115927 

33 


n 

4% 

^y^'/c 

5% 

5%% 

6% 

1 

1.00 

1.00 

1.00 

1.00 

1.00 

2 

.49019608 

.48899756 

.487804^8 

.48661800 

.48543689 

3 

.32034854 

.31877336 

.31720856 

.31565407 

.31410981 

4 

.23549005 

.23374365 

.2:3201183 

.23029449 

.22859149 

5 

.18462711 

.18279164 

.18097480 

.17917644 

.17739010 

6 

.15076190 

.14887839 

.14701747 

.14517895 

.1433026:? 

7 

.12660961 

.12470147 

.12281982 

.120964'}.2 

.11913502 

8 

.10852783 

.10660905 

.10472181 

.10286401 

.1010:3594 

9 

.0941.9299 

.09257447 

.09069008 

.08883946 

.08702224 

10 

.08329094 

,08137882 

.07950458 

.07760777 

.07586796 

11 

.07414904 

.07224818 

.07038889 

.06857065 

.06689294 

12 

.06655217 

.00406619 

.06282541 

.06102923 

.05927703 

13 

.06014:373 

.05S27535 

.0504.5577 

.05408426 

.0.5296011 

14 

.054;)6897 

.05282032 

.05102397 

.04927912 

.04758491 

15 

.04994110 

.04811381 

.04634229 

.041.62560 

.04296176 

16 

.04582000 

.04401537 

.04226991 

.04058254 

.0389.5214 

17 

.04219852 

.04041758 

.03809914 

.03704197 

.03544480 

18 

.03899333 

.03723690 

.03554622 

.0.3391992 

.03235654 

19 

.03613862 

.03440734 

.03274501 

.0311.5006 

.02962086 

20 

.03358173 

.03187614 

.03024259 

.02867933 

.02718456 

21 

.03128011 

.02960057 

.02799611 

.02646478 

.02500455 

22 

.02919881 

.02754546 

.02597051 

.02447123 

.02:304557 

23 

.02730906 

.02568249 

.0241:3682 

.02266965 

.02127848 

24 

.02558683 

.02398703 

.02247090 

.0210.3580 

.01967900 

25 

.02401196 

.02243903 

.02095246 

.01954935 

.01822672 

26 

.02256738 

.02102137 

.01956432 

.01819.307 

.01690435 

27 

.02123854 

.01971946 

.01829186 

.01695228 

.01509717 

28 

.02001298 

.01852081 

.01712253 

.01581440 

.014592.55 

29 

.01887993 

.01741-461 

.01004551 

.01470857 

.01357961 

30 

.01783010 

.01639154 

.01505144 

.01380539 

.01264891 

31 

.01685535 

.01544345 

.01413212 

.01291005 

.01179222 

32 

.01594859 

.01456:320 

.01328042 

.01209519 

.011002.34 

33 

.01510357 

.01374453 

.01249004 

.011.3:3469 

.01027293 

34 

.014:31477 

.01298191 

.01175545 

.01002958 

.00959843 

35 

.01357732 

.01227045 

.01107171 

.00997493 

.00897386 

36 

.01288688 

.01160578 

.01043440 

.009.36635 

.00839483 

37 

.0122:3957 

.01098402 

.00983979 

.00879993 

.007857-1:3 

38 

.01163192 

.010 W 169 

.00928423 

.00827217 

.007:35812 

39 

.01100083 

.00985567 

.00870462 

.00777991 

.00(i89377 

40 

.01052-349 

.00934315 

.00827816 

.00732034 

.00646154 

41 

.01001738 

.00886158 

.00782229 

.00689090 

.00605886 

42 

.00954020 

.00840868 

.00739471 

.00648927 

.00508342 

43 

.00908989 

.00798235 

.00099333 

.00611337 

.00,5:5:3:512 

44 

.00966454 

.00758071 

.00001025 

.00570128 

.00500606 

45 

.00820246 

.00720202 

.00026173 

.00543127 

.00470050 

46 

.00788205 

.00684471 

.00592820 

.00512175 

.004-tl485 

47 

.00752189 

.00050734 

.00501421 

.00483129 

.00414768 

48 

.00718005 

.00018858 

.0053184:3 

.00455854 

.0():5897()6 

49 

.00685712 

.00588722 

.00503965 

.004.30230 

.00306356 

50 

.00655020 

.00560215 

.00477674 

.00406145 

.00344429 

34 


n 

*% 

*y3% 

6% 

5V2% 

6% 

51 

.00625885 

.00533232 

.00452867 

.00383495 

.00323880 

52 

.00598212 

.00507679 

.00429450 

.00362186 

.00304616 

53 

.00571914 

.00483469 

.00407334 

.00342130 

.00286551 

54 

.0051.6912 

.00460519 

.00386438 

.00323245 

.00269602 

55 

.00523124. 

.00438754 

.00366686 

.00305458 

.00253696 

56 

.00500487 

.00418105 

.00348010 

.00288698 

.00238765 

57 

.00478932 

.00398506 

.00333032 

.00272900 

.00224744 

58 

.00458401 

.00379897 

.00313626 

.00258006 

.00211574 

59 

.00438836 

.00362221 

.00297802 

.00243959 

.00199200 

60 

.00420185 

.00345426 

.00282818 

.00230707 

.00187572 

61 

.00402398 

.00329461 

.00268627 

.00218202 

.00176642 

62 

.00385430 

.00314284 

.00255183 

.00206400 

.00166366 

63 

.00369237 

.00299848 

.00242442 

.0019.5258 

.00156703 

64 

.00353779 

.00286115 

.00230365 

.00184737 

.00147615 

65 

.00339019 

.00273047 

.00218915 

.00174800 

.00139066 

66 

.00324921 

.00260608 

.00208057 

.00165413 

.00131022 

67 

.00311451 

.00248765 

.00197757 

.00156544 

.00123453 

68 

.00298578 

.00237492 

.00187986 

.00148163 

.00116330 

69 

.00286272 

.00226745 

.00178715 

.00140242 

.00109625 

70 

.00274506 

.00216511 

.00169915 

.00132754 

.00103313 

71 

.00263254 

.00206759 

.00161563 

.00125675 

.00097370 

72 

.00252489 

.00197465 

.00153633 

.00118982 

.00091774 

73 

.00242190 

.00188606 

.00146103 

.00112652 

.00086504 

74 

.00232334 

.00180159 

.00138951 

.00106665 

.00081542 

75 

.00222900 

.00172104 

.00132161 

.00101002 

.00076867 

76 

.00213869 

.00164422 

.00125709 

.00095645 

.00072463 

77 

.00205221 

.00157094 

.00119580 

.00090577 

.00086315 

78 

.00196939 

.00150101 

.00113756 

.00085781 

.00064407 

79 

.00189006 

.00143434 

.00108222 

.00081243 

.00060724 

80 

.00181408 

.00137069 

.00102962 

.00076948 

.00057254 

81 

.00174127 

.00130995 

.00097963 

.00072884 

.00053984 

82 

.00167150 

.00125196 

.00093211 

.00069036 

.00050902 

83 

.00160463 

.00119663 

.00088694 

.00065395 

.00047998 

84 

.00154054 

.00114379 

.00084399 

.00061947 

.00045261 

85 

.00147909 

.00109334 

.00080316 

.00058683 

.00042681 

86 

.00142018 

.00104516 

.00076433 

.00055593 

.00040241 

87 

.00136370 

.00099915 

.00072740 

.00052667 

.00073956 

88 

.00130953 

.00095522 

.00069228 

.00049896 

.00035795 

89 

.00125758 

.00091235 

.00065888 

.00047273 

.00033757 

90 

.00120775 

.00087316 

.00062711 

.00044788 

.00031836 

91 

.00115995 

.00083486 

.00059689 

.00042435 

.00030025 

92 

.00111410 

.00079827 

.00056815 

.00040207 

.00028318 

93 

.00107012 

.00076331 

.00054080 

.00038096 

.00026708 

94 

.00102789 

.00072991 

.00051478 

.00036097 

.00025189 

95 

.00098738 

.00069799 

.00049003 

.00034204 

.00023758 

96 

.00094850 

.00066749 

.00046648 

.00032410 

.00024408 

97 

.00091119 

.00063834 

.00044407 

.0003071 1 

.00021135 

98 

.00087538 

.00061048 

.00042274 

.00029101 

.00019935 

99 

.00084100 

.00058385 

.00040245 

.00027577 

.00018803 

100 

.00080800 

.00055839 

.00038314 

.00026132 

.00017736 

35 


TABLE  VI 

Effective  Kate  Factors. 
The  following  are  the  effective  rate  factors  for  a  single 


mit. 

The  formula  is  jp= 

^p(Vl+i- 

-1) 

p 

2 

1% 

.00998756 

.01246118 

.01494417 

1%% 
.01742410 

2% 
.01990099 

4 

.00996272 

.01244183 

.01491636 

.01739631 

.01985173 

12 

.00995446 

.01242895 

.01489785 

.01736119 

.01981898 

P 
2 

2>/4% 

.02237484 

2Vb7c 
.02484567 

2%% 
.02731349 

.02977831 

31/2% 
.03469859 

4 

.02231261 

.02476899 

.02722087 

.02966829 

.03454978 

12 

.02227125 

.02471804 

.02715936 

.02959524 

.03445078 

P 
2 

4^^ 
.03960781 

^V2  7c 

.04450483 

5% 
.04939015 

.05426386 

6% 
.05921603 

4 

.03941363 

.04425996 

.04908894 

.05390070 

.05869538 

12 

.03928488 

.04409771 

.04888949 

.05366039 

.05841061 

The  effective  rate  factors  for  use  with  annuity  tables 
are  seldom  of  use  to  the  accountant.  The  factors  are  in- 
dicated by  the  fraction  ^-,  as  explained  on  pages  63  to  65, 

3  p 
where  these  factors  were  discussed.     In  cases  where  there 

is  an  effective  rate  and  the  yearly  rate  is  in  the  tables, 
these  rules  w^ill  apply : 

In  case  of  &„  or  a^,:  multiply  the  table  value  by  the 
yearly  rate  and  divide  by  th  correct  value  of  jp. 

1        1 

In  case  of  ^-  or  —  multiply  by  the  correct  value  of  j  p 

6  „       a , 

and  divide  by  the  yearly  rate. 


36 


TABLE  VII 
TEN-PLACE  LOGARITHMS  OF  THE  INTEREST 

RATIOS 


Rate  % 

1+i 

log. 

Rate  % 

1  +  i 

log. 

1  1/40 

1.01025 

.0044288591 

1/20 

1.0105 

.0045363179 

3/40 

1.01075 

.0046437500 

1/10 

1.011 

.0047511556 

1/8 

1.00125 

.0005425291 

1/8 

1.01125 

.0048585346 

3/20 

1.0015 

.0006509536 

3/20 

1.0ia5 

.0049658871 

7/40 

1.00175 

.0007593511 

7/40 

1.01175 

.0050732131 

1/5 

1.002 

.0008677215 

1/5 

1.012 

.0051805125 

9/40 

1.00225 

.0009760649 

9/40 

1.01225 

.0052877854 

1/4 

1.0025 

.0010843813 

1/4 

1.0125 

.0053950319 

11/40 

1.00275 

.0011926707 

11/40 

1.01275 

.0055022519 

3/10 

1.003 

.0013009330 

3/10 

1.013 

.0056094454 

13/40 

1.00325 

.0014091684 

13/40 

1.01325 

.0057166124 

7/20 

1.0035 

.0015173768 

7/20 

1.0135 

.0058237530 

3/8 

1.00375 

.0016255583 

3/8 

1.01375 

.0059308672 

2/5 

1.004 

.0017337128 

2/5 

1.014 

.0060379550 

17/40 

1.00425 

.0018418404 

17/40 

1.01425 

.0061450164 

9/20 

1.0045 

.0019499411 

9/20 

1.0145 

.0062520514 

19/40 

1.00475 

.0020580149 

19/40 

1.01475 

.0063590600 

1/2 

1.005 

.0021660618 

1/2 

1.015 

.0064660422 

21/40 

1.00525 

.0022740818 

21/40 

1.01525 

.0065729982 

11/20 

1.0055 

.0023820749 

11/20 

1.0055 

.0066799277 

23/40 

1.00575 

.0024900412 

23/40 

1.01575 

.0067868310 

3/5 

1.006 

.0025979807 

3/5 

1.016 

.0068937079 

6/8 

1.00625 

.0027058934 

5/8 

1.01625 

.0070005586 

13/20 

1.0065 

.0028137792 

13/20 

1.0165 

.0071073830 

27/40 

1.00675 

.0029216383 

27/40 

1.01675 

.0072141811 

7/10 

1.007 

.0030294706 

7/10 

1.017 

.0073209529 

29/40 

1.00725 

.0031372761 

29/40 

1.01725 

.0074276985 

3/4 

1.0075 

.0032450548 

3/4 

1.0175 

.0075344179 

31/40 

1.00775 

.0033528068 

31/40 

1.01775 

.0076411111 

4/5 

1.008 

.0034605321 

4/5 

1.018 

.0077477780 

33/40 

1.00825 

.0035682307 

33/40 

1.01825 

.0078544188 

17/20 

1.0085 

.0036759025 

17/20 

1.0185 

.0079610333 

7/8 

1.00875 

.0037835477 

7/8 

1.01875 

.0080676217 

9/10 

1.009 

.0038911662 

9/10 

1.019 

.0081741840 

37/40 

1.00925 

.0039987581 

37/40 

1.01925 

.0082807201 

19/20 

1.0095 

.0041063233 

19/20 

1.0195 

.0083872301 

39/40 

1.00975 

.0042138618 

39/40 

1.01975 

.0084937140 

1 

1.01 

.0043213738 

2 

1.02 

.0086001718 

37 


TABLE  VII— (Continued) 

TEN-PLACE   LOGARITHMS   OF  THE   INTEREST 

RATIOS— Continued 


Rate  % 

1  +i 

log. 

Rate  % 

1  +i 

log. 

2  1.40 

1.02025 

.0087066034 

3  1/20 

1.0305 

.0130479961 

1/20 

1.0205 

.0088130091 

1/10 

1.031 

.0132586653 

3/40 

1.02075 

.0089193886 

3/20 

1.0315 

.0134692323 

1/10 

1.021 

.0090257421 

1/5 

1.032 

0136796973 

1/8 

1,02125 

.0091320695 

1/4 

1.0325 

0138900603 

3/20 

1.0215 

.0092383710 

3/10 

1.033 

.0141003215 

7/40 

1.02175 

, 0093446464 

7/20 

1.0335 

.0143104810 

1/5 

1.022 

.0094508958 

2/5 

1.034 

.0145205388 

9/40 

1.02225 

.0095571192 

9/20 

1.0345 

.0147304950 

1/4 

1.0225 

.0096633167 

1/2 

1.035 

.0149403498 

11/40 

1.02275 

.0097694882 

3/10 

1.023 

.0098756337 

11/20 

1.0355 

0151501032 

13/40 

1.02325 

.0099817533 

3/5 

1.036 

.0153597554 

7/20 

1.0235 

.0100878470 

3/4 

1.0375 

.0159881054 

3/8 

1.02375 

.0101939148 

4/5 

1.038 

.0161973535 

2/5 

1.024 

.0102999566 

9/10 

1.039 

0166155476 

17/40 

1.02425 

.0104059726 

9/20 

1.0245 

.0106119627 

4 

1.04 

0170333393 

19/40 

1.02475 

.0106179270 

1/10 

1.041 

.0174507295 

1/2 

1.025  ' 

.0107238654 

1/4 

1.0425 

.0180760636 

3/10 

1.043 

.0182843084 

21/40 

1.02525 

.0108297780 

2/5 

1.044 

.0187004987 

11/20 

1.0255 

.0109356647 

1/2 

1.045 

.0191162904 

23/40 

1.02575 

.0110415256 

3/5 

1.026 

.0111473608 

3/5 

1.046 

.0195316845 

5/8 

1.02625 

.0112531701 

3/4 

1.0475 

.0201540316 

13/20 

1.0265 

.0113589537 

4/5 

1.048 

.0203612826 

27/40 

1.02675 

.0114647115 

9/10 

1.049 

.0207754882 

7/10 

1.027 

.0115704436 

29/40 

1.02725 

.0116761499 

5 

1.05 

0211892991 

3/4 

1.0275 

.0117818305 

1/2 

1.055 

.0232524596 

31/40 

1.02775 

.0118874854 

4/5 

1 .  028 

.0119931147 

6 

1.06 

.0253058653 

33/40 

1.02S25 

.0120987182 

1/2 

1.065 

.0273496078 

17/20 

1.0285 

.0122042960 

7/8 

1.02S75 

.0123098482 

7 

1.07 

.0293837777 

9/10 

1.029 

.0124153748 

1/2 

1.075 

.0314084643 

37/40 

1.02925 

.0125208757 

19/20 

1.0295 

.0126263510 

8 

1.08 

.0334237555 

39/40 

1.02975 

.0127318006 

3 

1.03 

.0128372247 

9 

1.09 

.0374264979 

10 

1.10 

.0413926852 

38 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 

This  book  is  DUE  on  the  last  date  stamped  below. 


SEP  1  4  1962 


SEP  i  ? 


Form  L9-50m-9.'60(B3610s4)444 


AA    001  020  219    0 


